**CONTINUATION OF IMMEDIATELY-PRECEDING BLOG ENTRY, # 48**

**How do the**

**N**

__Q__

*"*

*First*

__Dialectical__Meta

*-*

*Numbers*

*"***Emerge**

*Intuitively*from the**"**

**N**

**atural**

**"**

**Numbers**

*?*Dear Readers,

How can we best understand --

**and**

*intuitively***-- the emergence of the**

__dialectically__

__N__

__Q___**of the**

*system***N**

__Q__*"*as a

__dialectical__**meta**-**numbers**"*"*to the "<<

**first****-**__contra__**system**"**>>-**

*arche'*

*system**"*of the "

**atural" Numbers,**

__N__**N**...

**...**

__=__**{ 1, 2, 3, 4, . . . }**

*?*

That is, the first transition in the

**F**.

**.**

__E__**.**

*D*

*meta**-*

**system****of the**

*atic*__dialectic__**is --**

__dialectical__arithmetics

__)-|-(__**1**....

**=**....

__)-|-(__**0**

**^(**

**2**

**^**

**1**

**)**....

**=**....

__N__**..**

_____**+**..

__N__

__Q___How do we "justify" this transition, from

*system*

__N__**to**

_____

*system*

__N__**..**

_____**+**..

*system*

__N__**,**

__Q___**and**

*intuitively***?**

__dialectically__First, in what ways is

*system*

__N__**a**

__Q___**of**

__qual__itative opposite

*system*

__N__**, in the sense of a combined, "complementary opposite" and "supplementary opposite" of**

_____

*system*

__N__**?**

_____Well,

__N__**stands for an axioms-system of arithmetic for a "space" of "**

_____**atural Numbers", such that each of these "Numbers" can be characterized as an**

__N__*"*.

__unqual__ified**"**__quant__ifierThe symbol

__N__**, on the contrary, stands for an axioms-system of arithmetic for a "space" of**

__Q___*"*

**-**__meta__**atural**

__N__*Numbers", such that each of these*

**-**__meta__*"*Numbers" can be characterized as an

**-**__meta__*"*[

__unquant__ifiable**]**

*ontological**.*

**"**__qual__ifier[

__Note__: the phrase "

**three**<<

**>>" exemplifies a**

*species**"*, or a

**ified**__qual__**ifier"**__quant__*"*-- the

**ified**__quant__**ifier"**__qual__*"*,

**ifier"**__quant__**three**, is

*"*ified",

__qual__**, or in terms of**

*ontologically**"*

*kinds**"*[<<

**>>] of being, by the word "<<**

*gene***>>".**

*species*The phrase "

**three**

**inches**" also exemplifies a

*"*, or a

**ified**__qual__**ifier"**__quant__*"*-- the

**ified**__quant__**ifier"**__qual__*"*,

**ifier"**__quant__**three**, is

*"*ified",

__qual__**, or in terms of**

*metrically***[<<**

*units***>>] of**

*monads***ment, by "**

*measure***inches**".

The word "

**three**", by itself, exemplifies and

*"*.

**ified**__unqual__**ifier"**__quant__the word "<<

**>>, by itself, exemplifies an**

*species**"*

**ified**__unquant__

*ontological*

__qual__*ifier"*.

the word "

**inches**", by itself, exemplifies an

*"*

**ified**__unquant__

*metrical*

__qual__*ifier"*.]

But, how does

*system*

__N__**, this complementary/supplementary**

__Q___**of**

__opposite__

*system*

__N__**, emerge**

_____

__from__

*system*

__N__**?**

_____There is a

**-- indeed, an**

*conceptual ambiguity**"*,

**-**__intra__**duality**"*"*, or "

__internal__**duality**"

__self__*-*-- in the concept of "

**duality**"**atural Number"; in the "first-order", "Peanic"**

__N__**of the "**

*system***atural" Numbers,**

__N__

__N__**, that leaves it open to**

_____*"*Standard Models" of the "

**-**__Non__**atural" Numbers,**

__N__**N**, such as that of the

__N__

__Q___**.**

*system*This

*"*duality" is that of

**intra**-**--**

__card__inality versus__ord__inality

__card__inal numbers versus__ord__inal*"*.

**numbers**"The "

**atural Numbers" are supposed to be, "simultaneously", "inseparately", both**

__N__*"*and

**finite**"__card__inal numbers*"*.

**finite**"__ord__inal numbersBut note how the first four, "

**first**order" Peano Postulates, or "Peano Axioms", of the five Peano Postulates total ["

**first**" order because they "talk about" only individual "

**atural" numbers --**

__N__**about, e.g.,**

__not__**ities shared by ensembles of such individual numbers, such as "odd-ness" versus "even-ness", or "composite-ness" versus "prime-ness" ["**

__qual__**second**order"], and

**about qualities shared by**

__not__**such ensembles ["**

*ensembles of***third**order"], etc.], are all about

**,**

__ord__inality**about**

__not__**:**

__card__inalityPeano Postulate

**a**.:

**1**is a

**atural Number.**

__N__Peano Postulate

**b**.: The successor of any

**atural Number is also a**

__N__**atural Number.**

__N__Peano Postulate

**c**.: No two distinct

**atural Numbers have the same successor.**

__N__Peano Postulate

**d**.:

**1**is not the successor of any

**atural Number.**

__N__Now, although, on the one hand,

**can be reduced, by the "modern" [i.e., "capitalist"], law-of-value-permeated <<**

__card__inal numbers**>> of**

*mentalite'***, to the**

*human psychohistory**"*-- even given that

**purely**-**"**__quant__itative

__card__**is also standardly formulated as an attribute, a predicate, a**

*inality***of**

__qual__itative determination**[of**

*sets***elements] -- on the other hand,**

__qual__itative**, and**

__ord__inal numbers**in general, cannot be so reduced.**

__ord__inalityThere is a

**aspect to**

__qual__itative**, to**

__ord__inality

__ord__er*-*in a progression, although one can certainly

**position***"*through, e.g.,

**count out**"**n**categories, or

**n**

**systems, in a categori**

**progression, or system**

__es__**progression, to determine the**

__s__**of a given category, or system, in that progression, so that there is something**

__ord__inal value**about**

__quant__itative**as well.**

__ord__inal values*"*ness" is also a

**First**-**ity, though a rather abstract**

__qual__**ity.**

__qual__*"*ness" is also a

**-**__Second__**ity, a**

__qual__**of a term in a progression of terms/categories.**

__qual__itative determinationFor instance --

**a**....

**b**....

**c**....

**d**

**. . .**

-- conveys the same sense of

**as --**

__ord__inality**1**....

**2**....

**3**....

**4**

**. . .**

-- although the latter have also

**plicit**

__ex__**connotations, whereas the former do not.**

__card__inal numberEach of the former represent, not

**s, which,**

__card__inal number**plicitly, would have only**

__ex__

*purely**-*among them, but phonetic symbols -- "phonetic ideograms", or "phonograms" -- each one representing, albeit loosely, a distinct sound of speech, one

__quant__itative differences**, not**

__qual__itatively different**, from the "sound of speech" value of every other such symbol in the alphabet in question.**

*purely*__quant__itatively differentThat is, for the latter --

**1**

**<**

**2**

**<**

**3**

**<**

**4**

**<**

**. . .**

**-- but these relations do**

**hold for the former.**

__not__It is

**true that --**

__not__**a**

**<**

**b**

**<**

**c**

**<**

**d**

**<**

**. . .**

-- but, on the contrary, it

**true that --**

__is__**a**

**is**

**less than,**

__neither__**equal to,**

__nor__**greater than**

__nor__

**b**

**is**

**less than,**

__neither__**equal to,**

__nor__**greater than**

__nor__**c**is

**less than,**

__neither__**equal to,**

__nor__**greater than**

__nor__**d**

**. . .**

-- i.e., that --

**a**

**is**

__qual__itatively**equal to**

__un__**b**

**is**

__qual__itatively**equal to**

__un__

**c**

**is**

__qual__itatively**equal to**

__un__**d**

**. . .**.

So, I understand the

**, conceptual, emergence of**

__dialectical__

__N__**from**

__Q___

__N__**,**

_____**, as follows:**

*intuitively*The

__N__

_____**of arithmetic, taken to be,**

*system***plicitly, that of an arithmetic of**

__ex__*"*, has a suppressed,

**pure**,**ified**__unqual__**izers, or**__cardinal__**ifiers"**__quant__*"*, inner "Janus face"; a suppressed

**"**__im__plicitized*"*

**inner**

*dual**"*, which is an arithmetical axioms-system of

*"*

__un__**inal**

__card__

__ordinal__*izers"*, or

**.**

__ord__inal__qual__ifiersThe "reflexion" of

__N__**upon**

_____

__N__**itself, as the "reflection" of a human subject/agent, "holding**

_____

__N__**in mind", and reflecting upon**

_____

__N__**itself, in the terms of**

_____

__N__**itself, from the thus "mind-embodied" point-of-view and innate criteria**

_____

*of*

__N__**itself, produces a**

_____

__dialectical__*"*, or

**immanent critique**"*"*, of

**-**__self__**critique**"

__N__**itself,**

_____*"*

**by**"

__N__**itself, as "embodied" in the mind of that human subject/agent -- a complex human "mental operation", or human "mental process", which**

_____*"*,

**outs**"*"*,

**outers**"*"*, or

**-**__de__**suppresses**"*"*, that previously

**explicitizes**"**, previously**

*hidden**"*, previously

**occulted**"**"inner Janus face".**

*suppressed*This

**"mental process" process can be modeled**

*complex**"*, in "shorthand" mnemonic form, by the [mostly] symbolic-"ideogramic" expression [wherein '

**simply**"**---->**' denotes the phrase "goes to", "becomes", or "develops into"] --

__N__

_____**---->**

__N__

_____*"*

**of**"

__N__**....**

_____**=**

__N__

_____**(**

__N__

_____**)**....

**=**

NN

_____*"*

**times**"

__N__**....**

_____**=**

NN

_____*"*

**confronts**"

__N__**....**

_____**=**

NN

_____*"*

**interacts with**"

__N__**....**

_____**=**

NN

**"**

_____**critiques**"

__N__**....**

_____**=**

NN

**<<**

_____**>>**

*aufheben*

*s*

__N__**...**

_____**=**

NN

**x**

_____

__N__**....**

_____**=**

NN

_____*"*....

**squared**"**=**....

__N__

_____**^**

**2**....

**=**

NN

**...**

_____**+**...

**-**

__delta__

__N__**....**

_____**=**

NN

**...**

_____**+**...

__N__

__Q___-- wherein "

**-**

__delta__

__N__**" denotes a**

_____**-- and an increment of "new"**

__qual__itative increment

__idea__*-*-- to

**ontology**

__N__**, and wherein**

_____

__N__**denotes the**

__Q___**,**

__speci__fic**,**

__ex__plicit**of the hidden, suppressed,**

__ex__ternalized form**plicit**

__im__**ternal dual within**

__in__

__N__**.**

_____Thus --

__arithmetic____of__.....x.....

**inal "pure"-***card***ifiers***quant***........**

__itself__**=**

__arithmetic____of__.....

**inal "pure"-***card***ifiers***quant***+**.....

__arithmetic____of__

*non**-*cardinal**inal***ord***ifiers***qual*-- whereby the

__arithmetic____of__both

*non**-*cardinal**inal***ord***ifiers***qual***itatively**

__qual__**, and also**

*opposes***, the <<**

*supplements***>>**

*arche'*

__arithmetic____of__.

**inal "pure"-***card***ifiers***quant*In the context of

__N__**..**

_____**---->**..

__N__

_____**(**

__N__

_____**)**........

**=**........

__N__**.....**

_____**+**.....

__N__**, we can comprehend the <<**

__Q___**>> of the**

*arithmos*

__N__

__Q__*"*/

**meta**-**numbers**"*"*as arising from an <<

__ord__inal**"**__qual__ifiers**>> operation of the <<**

*aufheben***>>**

*arithmos***N**, upon the <<

**>>**

*arithmos***N**, i.e., as a "primordial" [

*self**-*]

*"*<<

**meta**-**>>**

*monad**-*of the

**ization**"**N**.

For a possible, but

**ical-process -- perhaps therefore easier to visualize -- one partially**

__phys__**to this "purely-**

__parallel__

__concept__*ual"*-process, consider the early, interstellar '''atomic clouds''', that were destined later to became "molecular clouds".

As the rate of stellar production -- via stellar nucleosynthesis, and via other stellar processes -- of higher atomic

*physio**-*<<

**>> surged-up, driving rising rates of influx of the**

*species***atom**products of these stellar processes into these inter-stellar clouds, local densities -- local physical-spatial concentrations -- of

**atoms**were achieved that were sufficient to give rise to the first relatively stable

**ever seen in our**

__molecules__**.**

*cosmos*The process of the emergence of

**, as <<**

__molecules__**>> of a**

*arithmoi*

*new***, or <<**

*kind***>>, of <<**

*genos***>>, or of**

*monads***, from out of these self-convergences, and self-concentrations, of**

*units***as the <<**

__atoms__**>> of their predecessor**

*arithmoi***, or <<**

*kind***>>, of <<**

*genos***>>, or**

*monads***, can be grasped as a mutual**

*units**"*, and as a mutual

**self**-**subsumption**"*"*, of the

**self**-**internalization**"**fic kinds of sub-<<**

*speci***>> of**

*arithmoi***that form**

__atoms__**fic kinds of**

__speci__**.**

__molecules__Each such

**molecule**<<

**>>, or**

*monad***, can be seen as the result of a**

*unit**self-*<<

**>>**

*aufheben***of a [usually]**

*mutual internalization***geneous multiplicity of**

__hetero__**atom**[sub-]<<

**>>, or [sub-]**

*monads***.**

*units*The

*'''self-re-entry'''*of atoms to form molecules; the coming together of these

**as**

__atoms__**superseded, for their locations, the former <<**

__molecules__**>>-ic**

*monad**"*ternity", or

__in__*"*-side", and the former <<

__in__**>>-ic**

*monad**"*ternity", or

__ex__*"*-side", creating a new [meta-]<<

__out__**>>-ic**

*monad**"*ternity", or

__in__*"*-side", and a new [meta-]<<

__in__**>>-ic**

*monad**"*ternity", or

__ex__*"*-side", plus, in that very process, creating a new

__out__*"*

**-**__qual__o**fractal**"**, or**

*scale***: that of the new**

*level***molecular**<<

**>> /**

*monads***, in which their predecessor,**

*units***atomic**<<

**>> /**

*monads***were <<**

*units***>>**

*aufheben**-*, <<

**conserved****>>-**

*aufheben*

*determinately**-*, and <<

**negated****>**

*aufheben**-*in terms of their

**elevated***"*

**-**__qual__o**fractal**"**/**

*scale***.**

*level**'''*

*Self-re-entry*

*'''*creates a new self.

*"Self-internalization"*of existing [ideo-]ontology creates new [ideo-]ontology.

"[

*Self-*]*Meta-*<<**>>***monad**-ization"*of existing <<**>> creates a***monads***of <<***new kind***>>.***arithmos*"[

*Self-*]*subsumption"*of existing [ideo-]ontology creates new [ideo-]ontology."[

*Self-*]*demotion"*[cf. Hegel] of existing [ideo-]ontology immediately gives birth to/promotesnew [ideo-]ontology.

Likewise, it is the human-mental "gathering-together" of the -- in this "prim

Thus, the

That is, "

....................<<

Likewise, it is the human-mental "gathering-together" of the -- in this "prim

**ial" case,**__ord__**geneous,**__homo____not__**genous --**__hetero__*"*(s)", or <<__unit__**(s)>>, of the**__monad__**N**"Peano**atural Numbers" -- namely, the**__N__**/<<**__unit__**>> denoted by**__monad__**1**-- into distinct multiplicities, as*"*<<**inal**__ord__**>>***species*__denomin__ators*"*; as the <<**>>-"contained"/-***aufheben**conserved**"*side" of the resulting new**-**__in__*"*, and made visible in and as these**meta**-**numerals**"*"*<<**inal**__ord__**>>***species*__denomin__ators*"*, that**the**__creates__**N****<<**__Q__**>>***genos***for**__qual__ifier*"***-**__ord__inality**in**-__gene__ral*"*[given the**,**__ord__inal__not__**, focus of the "first order" Peano Postulates], namely,**__card__inal**, the**__q__*"*of**"**__qual__itative__numer__ator**-- of**__all____each__**-- of the**__one__**N****, as the**__Q__*"*side" of each**-**__out__**N**__Q__*"*, which could equally-well be notated as**meta**-**numeral**"**, denoting**__n____ord__*inal-*__n__*umber-in-*__gene__*ral*, which, as a*"*ization" of the**-**__character__*"*tension" of the__in__*"*tension" of all Peano "__ex__**atural" numbers, must be a**__N__**,**__qual__ifier**a**__not__**, and which, again, given the**__quant__ifier**essence of the "first order" Peano Postulates, must stand for the**__ord__inal**ric**__gene__**of**__qual__ity**,**__ord__inal number**for the**__not__**ric**__gene__**of**__qual__ity**.**__card__inal numberThus, the

__q__**/n**, or the__n__**/n**, which constitute the**N****"space" of the**__Q____N__**system of arithmetic, arise "by", or "as", a human-mind-enacted, "self", or "mutual,**__Q_____sub__sumption*"*, and a "self", or "mutual,__in__ternalization*"*, of the**N****(s)/**__unit__**(s),**__monad__**1**, and, thus, by/as a human-mind-enacted, "self", or "mutual",*"*of the**meta**-**-**__unit__**ization**"**N****(s), "**__unit__**1**".That is, "

**", or "**__q__**", is a "self, or mutual,**__n____sub__sumption*"*, a "self, or mutual,__in__ternalization*"*, and a "self, or mutual,*meta**-*, of each**-**__unit__**ization**"**fic**__speci____sub__*-*<<**>> of***arithmos***1**s, in a way similar to, or analogous to, the way in which each given**molecule**is a "self, or mutual,__sub__sumption*"*, a "self, or mutual,*internalization**"*, and a "self, or mutual,*meta**-*, of a**-**__unit__**ization**"**fic sub-<<**__speci__**>> of***arithmos***atoms**, such that --....................<<

**>>.........<<***genos***>>...........<<***genos***>>.......................***genos***N****...**__Q____=__**...****{**...**_______________**,..**________________**,..**________________**,**. . .****}**..__=__**.................<<**

**>>***species***1**...<<**>>***species***2**....<<**>>***species***3**.................

**....**

**................**

__q__

__n__**{**.

**____**.

**}**..

__=__**..**

**{**.

**_____**.

**}**..

__=__**.**

....

**n**................**n****......**

**.............**

__q__**.................**

__q__**...................................**

__q__**.............**

__n__**..................**

__n__

__n__**{**..

**____**,....

**________**,....

**_________**,

**. . .**

**}**.....

__=__**.....**

**{**....

**____**,....

**________**,....

**_________**,

**... }**..

__=__**......**

**1**.........

**1**.

**+**

**.**

**1**........

**1**.

**+**

**.**

**1**

**.**

**+**

**.**

**1**............................

**1**..........

**1**.

**+**

**.**

**1**.......

**1**.

**+**

**.**

**1**

**.**

**+**

**.**

**1**.

**{**

__q__**1**,

__q__**2**,

__q__**3**,

**. . . }**......

__=__**.......**

**{**,

__q__|1**,**

__q__|2**,**

__q__|3**. . . }**.......

__=__**{**,

__q__/1**,**

__q__/2**,**

__q__/3**. . . }**.

The

*"*

**division**"*operations*invoked above -- and the

*"*-signifying

**division**"*"*used above -- should be seen as invoking a

**division**bars"**ralized,**

__gene__

*purely**-*,

__qual__itative

__non__*-*amalgamative,

**reducible kind of**

__ir__*"*that is native to

**division**"*"*

**-**__qual__o**-al**__fract__

__fract__*-tions"*, and to their

*"*, as signifying

**division**bar"*"*in the sense of Platonian-

**division**"**<<**

__dialectical__**>>, wherein their <<**

*diairesis***>>-**

*genos**"*, and their <<

**"**__numer__ators**>>-**

*species**"*, are mutually

**"**__denomin__ators**,**

__qual__itatively different

__not__**, and where the**

__quant__itatively different*"*side" of the <<

**-**__out__**>>-ic entity so "depicted", as a "**

__monad__**[y]", by the**

__unit__*"*, is also shown, again, by showing the

**ator"**__numer__*"*side" of itself, via the

**-**__in__*"*, but, there, in the guise of a

**"**__denomin__ator*"*; of its

**"**__multi__plicity*"*into the [typically

**division**"**geneous] multiplicity of its component, "internal", "implicit", <<**

*hetero***>>-**

*aufheben**"*<<

**contained**"/-**conserved****>>-[**

*species**sub-*]<<

**>>, or of its <<**

__monads__**>>-[**

*species**sub-*]

**.**

__units__In the

**al, "prim**

__speci__**ial" case of the**

__ord__

__q__**/n**, the

__q__**/n**

*"*

**-**__qual__o**-al**__fract__

__fract__*-tions"*do

**"amalgamate", or otherwise "reduce",**

__not__**because their**

__not__**,**

__numer__ator**, and their**

__q__**,**

__denomin__ator**n**in

**N**,

*represent pure arithmetical*

__both__**for**

__qual__ifiers**, as**

__qual__itatively distinct ontological__qual__ities**the case for a whole**

__is__

__dialectical__*-*-system

**arithmetical**

__s__

__sub__*-*of the

**progression****F**.

**.**

__E__**.**

*D*

*meta**-*

**systematic**-__dialectical__**of the**

*progression*

__dialectical__**, starting with --**

*arithmetics*

__N__

__q__**/BA [----)**

__q__/24-- which we have exposited here in an earlier entry to this blog.

**, in the**__No__**al case of the**__speci____q__**/n**, their**,***universal*__numer__ator**, and the**__q__*particular***of each,**__denomin__ator**rically given by the "***gene***atural"**__N__**umber variable**__n__**n**, do**"amalgamate", or otherwise "reduce", because their**__not__**is the**__numer__ator*"***purely**-**"**__qual__itative*gene***[standing for the***ric*__ord__inality__qual__ifier**of**__qual__ity*"*, or of "<<**-**__ord__inality**in**-**general**"**>>***genos*__ord__inality*"*], whereas their**are**__denomin__ators*"*"**purely**-**"**__quant__itative**atural"**__N__**umber**__n__**, so that**__quant__ifiers**and**__numer__ator__denomin__ator**differ**__do__**, i.e., because**__qual__itatively**and**__numera__tor**are**__denomin__ator*"*, but**mutually****"**__hetero__geneous**in this most rudimentary sense, a sense which is "prim**__only__**ial" with respect to the**__ord__**/**__numer__ator**characteristic of the rest of the progression of the**__denomin__ator__qual__itative*differences***F**.**.**__E__**. systems of***D***: numerator and denominator differ**__dialectical__arithmetic**itatively in this limiting/extreme case because the denominator is purely-**__qual__**itative, while the numerator is purely-**__quant__**itative.**__qual__This "rudimentarity", or "prim

It is only

The

**iality", arises because**__ord____N__**is only the**__Q___**system of arithmetic -- and the first***second***plicitly**__ex__**such system -- after the <<**__dialectical__**>> system,***arche'*__N__**, in the**_____**F**.**.**__E__**.***D**meta**-***systematic**-__dialectical__**of the systems of***progression***.**__dialectical__arithmeticIt is only

__q__**|1**, or__q__**/1**, or__q__**1**that does**have an <<**__not__**>> of "two or more"***arithmos***N****/<<**__units__**>> --"**__monads__**1**s"-- as its*"*side" /**-**__in__**/**__denomin__ator**.**__sub__scriptThe

__q__**/1***"*has only a**meta**-**numeral**"*"*<<**-**__non__**>>, a single***arithmos***1****/<<**__unit__**>>, as its**__monad__*"*side" /

**-**__in__**/**

__denomin__ator**.**

__sub__scriptThus, as expected, the <<

**>> differs from, and is**

*arche'***in relation to, all of its**

*unique*[

The intuitive account of the bifurcation of

The

Not surprisingly, from a

Indeed, if we denote the axioms-system, or "rules-system", of original Boolean logic-arithmetic/logic-algebra, by

*"*] successors / descendants.**-**__Qual__o**Peanic**"__Algebras of Logic, and Arithmetics of Logic, Boolean versus____"__**Contra**-**Boolean**"**.**__[__*Dialectical*]The intuitive account of the bifurcation of

__N__**into**_______N_______**+**__N__**set forth above is a special case of a general account of the human**__Q___*"*modeled by the arithmetical and algebraic operations of the**mental operations**"__N____Q___*"*, and its**contra**-**Boolean**"__arithmetic__*"*.**contra**-**Boolean**"__algebra__The

**F**.**.**__E__**. writings often characterize, in particular, the**__D____N__**arithmetic, as a**__Q___*"*[of/for a**contra**-**Boolean**__arithmetic____dialectical__**]", which gives rise to a***logic**"*.**contra**-**Boolean**"__algebra__Not surprisingly, from a

**point-of-view -- given the "connexion-by-opposition" between the original "Boolean arithmetic/algebra", and the**__dialectical__*"*of the**contra**-**Boolean**__arithmetic__**/**"__algebra____N__**system, there is a deep resonance, including a sharp difference, between Boole's account of the algorithmic operations of the original arithmetic/algebra that bears his name -- which he presented, beginning in his**__Q___**1847**work**, as a symbolic model of the human "mental operations" of [<<**__The Mathematical Analysis of Logic__**>>,***verstand*__sub__*-*] logical thought -- and of the generic version of the account of the algorithmic operations of the__dialectical____N__**arithmetic/algebra, as also an ideogramic-symbolic simulation of the human "mental operations" of <<**__Q___**>>,***vernunft*__dialectical__*thought.*__-__**logical**Indeed, if we denote the axioms-system, or "rules-system", of original Boolean logic-arithmetic/logic-algebra, by

__W____E__**, taking it to be the <<**_____**>> of all arithmetics/algebras of/for logic, and per a mnemonic which we will soon explain, then, in effect, in a different --***arche'*__unit__*-*interval-restricted -- "mutually perpendicular" direction/dimension of*axioms**-*from that of the**systems progression****F**.**.**__E__**.***D**meta**-***systematic**-__dialectical__**of the systems of***progression***, we can model the bifurcation of the Boolean arithmetic/algebra of logic,**__dialectical__arithmetic__W____E__**, into**_______W____E_______**+**__W__**, using the same generic symbolical-algorithmic process that we used, above, to model the bifurcation of**__Q_____N__**into**_______N_______**+**__N__**, so that**__Q_____W__**is grasped as an**__Q___**to**__opposite____W____E__**, in the sense of a**_____*"*, or of a**complementary****supplement**"*"*, to**supplementary complement**"__W____E__**--**_______W____E__**..**_____**---->**..__W____E_______**(**__W____E_______**)**........**=**........__W____E__**.....**_____**+**.....__W__**.......................**__Q___**[----)**.......................

__q__**/1**..

**---->**..

__q__**/1**..x..

__q__**/1**........

**=**........

__q__**/1**.....

**+**.....

__q__**/2**

Recall that

**W**, the "space", or "set" of the "

**hole" numbers, is defined herein as that of --**

__W__**{ 0**,

**1**,

**2**,

**3**,

**. . .}**

-- and we can also grasp the "[analytical"] geometry" of

__W__**as an <<**__Q___**>>***aufheben**"*<<

**meta**-**>>**

__monad__*-*, or

**ization**"*"*, of the

**meta**-**-**__unit__**ization**"

*unit**-*interval-restricted,

*unit**-*interval interior "elided", "[analytical"] geometry" of Boole's

__W__

__E__**, the Boolean "[analytical"] geometry" being that of an "end-points only" [**

_____**plicitly, at least] single**

__ex__**interval --**

*unit***|<----**

**0**

**B**

**/**

**0**

**B**

**-----**

**>|**

**0**

**B**..........................................

**1**

**B**

**............**

**<<<**

**1**

**B**

**/0**

**B**

**.**

**..............................**

**.**

-- wherein

**1**

**B**denotes "The Universe" [the class of

**individuals/<<**

__all__**>>] and wherein denotes "Nothing" [the class that is**

*monads***of all individuals / <<**

__empty__**>>].**

*monads*So, per Boole, in his

**1847**

**, and in his**

__The Mathematical Analysis of Logic__**1854**

**, Boolean-algebraic expressions/equations model human "mental operations" via symbols -- via ideograms, and via [e.g., mnemonic] phonograms converted for use as ideograms.**

__Laws of Thought__The generic mental operation -- represented each of by Boole's "logical ideograms" that are also "class" ideograms, or "class operators" -- is the operation that Boole calls "

**lection" [or, sometimes,**

__E__*"*election"].

__s__It is also for this reason that we have characterized Boole's

*unit**-*interval "arithmetic of/for logic", as "The

**lector Arithmetic", and that we have symbolized his axioms-system by**

__E__

__W__

__E__**.**

_____For instance, take

**w**as a Boolean "class ideogram" representing, by

*"*tension", the

__in__**ity of "**

__qual__**w**hite[-ness]", and, by

*"*tension", the "class" of all "

__ex__**w**hite things" [but still, in Boole's algebra, always as a [members-

**plicit]**

__im__*"*,

**tension"**__in__**as an**

__not__**plicit "**

__ex__**tension", or "set", of**

__ex__**plicitly-**

__ex__**"elements", or "members"], and take**

*listed***b**as a Boolean "class ideogram" representing the "class" of "

**b**irds".

Then, with those "givens" given, the Boolean expression

**w**

**x**

**b**, or, simply,

**wb**, symbolizes the "

**lection", by the "class"**

__E__**w**, from out of the "class"

**b**, of all of

**w**'s "likes"; of all "things", or "individuals", "within"

**b**, that "possess" the

**ity denoted by**

__qual__**w**, i.e., the "

**lection" of all "**

__E__**w**hite things" from out of the "class" of all "

**b**ird-things", and, thereby, the formation of a new "class", of "

**w**hite-

**b**ird-things", as the result, or "product", of this

*symbolically-*simulated,

*"logical-multiplication"-*simulated, "mental operation".

That is,

**wb**represents the "class" of "

**w**hite

**b**irds", which would be the

**set of the corresponding sets, or**

*intersection**"*tensions",

__ex__**W**and

**B**.

Likewise,

**bw**represents the "

**lection", by the**

__E__**b**s from out of the "class" of all of the

**w**s, of all of the

**b**s that are already among the

**w**s, or are already "in"

**w**, the result of this "logical multiplication" of

**b**and

**w**, again being the "class" of "

**b**ird things that are also

**w**hite-things", which is equivalent to the class

**wb**--

**bw**

**=**

**wb**

**-- so that Boolean "logical multiplication", or his "**

**lection" operation, is "commutative".**

__E__Now let us consider, in this light, the Boolean equation which Boole names

*"*[

**The Fundamental Law of**

__sub__*-*, <<

__dialectical__**>>]**

*verstand*

*Thought**"*, or

*"*[exo-]

**The Law of**

*Duality**"*, namely --

**x**x

**x**

**=**

**x**

-- or --

**x^2**

**=**

**x^1**

**=**

**x**

-- or simply --

**xx**

**=**

**x**

in relation to what

**F**.

**.**

__E__**. names**

__D__*"*, or

**the fundamental law of**"__dialectical__thought*"*

**the law of**

__self__*-*

**duality***"*, namely [using the underscores, as always in my blog entries here, to connote the

*"*Boolean-ness" of the operators so underscored] --

**-**__contra__**x**

__x__

__x__**=**

__x__**+**

**-**

__delta__

__x__-- or --

__x__**^2**

**=**

__x__**+**

**-**

__delta__

__x__-- or, simply --

__xx__

**=**

__x__**+**

**-**

__delta__

__x__**wherein**

**-**

__delta__**symbolizes the irruption of an incremental qualitative, categorial**

__x__**of**

__unit__**["ideo-", and/or "physi[c]o-"]**

*unprecedented new***, as an outcome, as a result, or as a "product", of the**

*ontology**"*operation", of the

**-**__self__*"*application", or

**-**__self__*"*reflexion" of an old, already existent, already extant, or, at least, already possible, categorial

**-**__self__**,**

__unit__**.**

__x__Per

**F**.

**.**

__E__**., this**

__D__**-**

__delta__

__x__*"*of categorial ontology, will be the result of a specific

**purely**-__qual__itative**categorial unit****"**__increment__*"*<<

**self**-**>>**

*aufheben*

*operation**"*of

**upon**

__x__**itself, which will, typically, if not always, be representable as a**

__x__*"*, or as a

**meta**-**-**__unit__**ization**"*"*<<

**meta**-**>>**

*monad**-*, of the

**ization**"**, or <<**

__units__**>>, held**

*monads***plicitly "within" the**

__im__

*categorial**-*symbol, or "<<

__unit__**>>-as-a-**

*arithmos***y" symbol,**

__unit__**.**

__x__In Boole's "intuitive" interpretation of the Boolean expression,

**xx**, for the "logical multiplication" of

**x**by

**x**, the expression

**xx**connotes the "

**lection",**

__E__**out of the [Right-Hand, or RH] "class"**

__from__**x**, of all of the "things", or "logical individuals",

**plicitly "contained in" [the Left-Hand, or LH]**

__im__**x**, i.e., that each exhibit the

**that all members of "class"**

__qual__ity**x**share; the

**that defines "class"**

__qual__ity**x**,

**the LH "class operator", also**

__by__**x**.

The LH

**x**"

**lects" only and all**

__E__**x**s from out of the RH

**x**, and puts them all into the results, or "product", "class" on the RH side of the "logical equation"

**xx**

**=**

**x**.

Thus, indeed, per Boole's "intuitive" interpretation of these symbolic operations,

**xx**

**=**

**x**.

"

*logical**,*

__non__**linearity**"**x^2**,

**immediately to**

__reduces__*"*,

**logical linearity**"**x^1**

**=**

**x**:

**xx**

**=**

**x^2**

**=**

**x^1**

**=**

**x**.

Boole's

*"*[

**Fundamental Law of****]**

*linear*

*Thought**"*logical equation models a

*"*, i.e., a

**"**__simple__reproduction of classes*"*, by means of the "self-

__simple__**reproduction**"**of ideas****lection" "mental operation" of the human mind --**

__E__

**idea**

**n**

**x**

**idea**

**n**

**=**

**idea**

**n**

-- depicting a

**universe of human knowledge, and thus implicitly denying all possibility of an**

__static__*"*

**"**__expanded__reproduction of ideas--

__idea__**n**

**x**

__idea__**n**

**=**

__idea__**n**

**+**

__idea__**n**

**+**

**n**

-- modeling an

__ir__reducibly*"*,

**"**__non__linear**, and**

*dynamical**"*[i.e., a

**-**__meta__**dynamical**"*, or [*

**olutionary**__rev__**]**

*a*

*periodically*__self__*-*], universe of human knowledge.

__rev__olutionizingThus, Boole's "fundamental" mental operation of "self-

**lection", symbolically simulated by his algebraic "logical equation"**

__E__**xx**

**=**

**x**, as mirrored by the self-multiplication of the

**of the**

*two values***oolean arithmetic,**

__B__**0**

**B**and

**1**

**B**, i.e. --

**0**

**B**x

**0**

**B**

**=**

**0**

**B**, mimicking ordinary

**W**-arithmetic's

**0**

**x**

**0**

**=**

**0**

-- and --

**1**

**B**x

**1**

**B**

**=**

**1**

**B**, mimicking ordinary

**W**-arithmetic's

**1**

**x**

**1**

**=**

**1**

-- with the "space" of "Boolean arithmetic" defined as follows --

**W**

__E__

__=__**{ 0**

**B**,

**1**

**B**

**}**

-- arithmetically models

**.**

*formal logic*Please note again that

**1**

**B**connotes

*"*in Boole's arithmetic of logic, and that

**-**__Every__**thing**"**0**

**B**connotes

*"*. Therefore, Boole uses the expression

**-**__No__**thing**"**(**

**1**

**B**

**-**

**x**

**)**, interpreted as meaning

*"*

**Universe**

__minus__**x**

*"*, to connote

*"*those things "contained in"

**-**__Every__**thing**__but__

**x**

*"*, or

*"*

**in the**__All__things**Universe**

__except__those that are**x**

*s"*, i.e.,

**(**

**1**

**B**

**-**

**x**

**)**connotes the "complement" of

**x**, or the "complementary

__opposite__*"*of

**x**.

Indeed, Boole adopts

**xx**

**=**

**x**as his

*"*at least partly because it represents a

**fundamental equation of****formal**-**logical****thought**"**fication, in a single expression, of two of the traditional "laws" of**

__uni__

*Aristotelian*

*formal***, in that the two modes of its "algebraic decomposition", yield those two, traditionally "separate",**

*logic**"*, or "rules".

**laws**"First, for the "multiplicative decomposition", or "factoring", of an algebraic "implicand" equation of the equation

**xx**

**=**

**x**[using '

**==>**' to denote the word "implies"] --

**x**

**=**

**xx ==>**

x

x

**-**

**xx**

**=**

**xx**

**-**

**xx ==>**

**x**

**-**

**xx**

**=**

**0**

**B**,

**==>**

**x(**

**1**

**B**

**-**

**x)**

**=**

**0**

**B**

-- which is the Boolean formulation of the traditional

*"*, meaning

**law of**-__non__**contradiction**"*"those*

**things**which exhibit both the quality**x**

*and the quality*

**-**__not__**x**

*are*.

**-**__No__**things**"**TO BE**

**CONTINUED NEXT BLOG ENTRY**

Regards,

Miguel

## No comments:

## Post a Comment