__Full Title__:

*On the Nature of the Opposition between*

*the*

__N___

*'*

**First**

*Standard*

*Arithmetic**'*

**and**

**the**

_{N}

__Q___

*'*

**First**__Dialectical__

**Arithmetic***':*

"Section

**2**" --

**Fruition of the**

__N___**~**

_{N}

__Q___

*Opposition*in a__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**Dear Readers,

Below, I have reproduced, for your cognitive pleasure, the middle part, "section

**2**", of the forthcoming

**F**.

__.__

**E**__. Vignette__

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

**Foundation**, which I have also added to the original blog-entry of this title, dated 14 November 2013, URL: http://feddialectics-miguel.blogspot.com/2013/11/on-narture-of-opposition-between-first.html.

I have reproduced it here, also, as a separate blog-entry, so as to alert you to this update / addition of content to the original blog-entry of this title.

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****Fruition of the**

__N___**~**

_{N}

__Q___

*Opposition*in a__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

These [

On the contrary, these [

These

**ric***gene**'*, or**category**-**level****unit**'*'*as a**category****/«***unit***»', each***monad*__q__**stands,**_{n}**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, we hold, logically,

**-- 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***/ '''[de-]***operation**fl*,**xion'''**__e__**fic to the***speci*_{N}**system/operator together with the**__Q___**system/operator, in their**__N___**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,

*intuitively*, gives rise, in the perceiving human mind, to a new*"*, i.e., to a new, third, idea / system; to a new**kind**"*'*, 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__opposition
This
human-mental process, whereby the mind of the 'presentor', and, if the
presentation/'presentor' is successful, also the minds of the
'presentees' -- all constituting the human subjects/agents who are
willingly/'will-fully' conducting these mental processes --
'mentally-embody', and 'mentally simulate', this antithesis, this
opposition of the outward meaning of

These new, higher,

Therein, the

Together,

The '

_{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,

*'*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 qualifier', written [approximately] as__**Mo****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 category itself --**__not__**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 generic [relative]***qual*__base__*-***level****[**__individual__*sub**-*]**/ [**__u__*nit**sub**-*]«**»***monad***relative to the****,**__category____q__**’, as their [**_{n}**-**__meta__**]****/[***unit*__meta__*-*]«**», by***monad**fying the quantitative determination, the cardinal quantity determination,***speci****u****, of the**_{n}__nits-__**u****ifier***qual**fier/determination,***speci**__u__^{o}**.**_{n}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 dialectical arithmetic, given it axioms.Instead, 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 step of the presentation, with**_{n}**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 'ideogramic' shorthand -- e.g., 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**", 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*__", as well as a '__**pounds****metrical****ifier', in this case "***quant***three**" -- all three components /*'''factors'''*-- all appear at once / all occur '''multiplied''' together, per our**ralized concept of [generally***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**unit", of**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 measured in terms of the 'metrical «_{1}(t) )**»-***monad***ifier' of***qual***ength,**__L__**[**, say, measured, again, in the__L__]*unit**-*of-measure**itative metrical «***qual***», or unit, of the "***monad***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 measured in terms of the 'metrical «_{1})**»-***monad***ifier' of***qual***ass,**__M__**[**, say, measured, again, in the__M__]*unit**-*of-measure**itative metrical «***qual***» of the "***monad***pound**", and whose "**ontrol[ parameter-]space" -- in this case, 'masses-space' -- directional unit-vector for "particle"**__c__**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**dynamical systems theory*total-differential equation [and, thus, typically also for a

*"*entailing, and therefore also typically

**singularity**"-*]dynamical mathematical model.*

**meta**-Note that each of these first two, "

**phase**space", classical "

**state**-space", examples actually each require not three but four '''factors''',

*'''*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 ‘generalized-multiplicative’ combination/conjunction with***quant**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__*-***-***units***level****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 to their dynamical versions, e.g., to their**_{n}*'''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*
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