**The**<<

*of*__Dialectic__**>>**

*Arithmos***and**<<

**>>**

*Monad*For definitions of key terms used below -- <<

**>>, <<***arithmos***>>, <<***monad***>>, "***arche'***meta**-**monadization**" -- see:Dear Readers,

According to Louis Couturat [and I agree], Leibniz's youthful treatise

*"*laid the groundwork for his later proposals for a universal-encyclopedia-based universal algebra, or "character-language", of all human thought, his <<**On The Art of Combinations**"**>>, whose actualization encountered formidable obstacles which Leibniz, his contemporaries, and his epigones were all unable to overcome --***Characteristica Universalis***F**.

**.**

__E__**. holds that the keys to the practicable actualization of such a <<**

__D__**>> are:**

*Characteristica Universalis***(1)**The realization of the [ideo- and physio-]ontological

**of the**

__fecundity__*'*combinations' of the <<

**-**__self__**>> ontological category, or term, of a**

*arche'***, or**

*categorial progression*

*terms**-*-- as opposed to

**progression****the ontological**

*presuming***of such**

*sterility**'*combinations' [

**-**__self__**Leibniz's and Boole's ontologically-reductionist and Parmenidean "fundamental law of thought",**

*a la***xx**

**=**

**x**, postulated by them for every ontological category/term

**x**], and;

**(2)**Interpretation of such fecund

*'*combinations' in terms of the

**-**__self__*'*[

*self**-*]

*meta**-*<<

**>>-**

*monad*

*ization**'*of the <<

**>> of each predecessor self-hybrid <<**

*monad*__s__**>>-of-<<**

*arithmos***>>, as the name/description of the process whereby the <<**

*monad*__s__**>> of its successor self-hybrid <<**

*monads***>>-of-<<**

*arithmos***>> -- in such categories/<<**

*monad*__s__**>>**

*arithmoi***-- are constructed.**

*progressions*These keys to the

**F**.**.**__E__**. breakthrough to a**__D__**<<**__dialectical__**>> are slated to be detailed in a future entry to this blog.***Characteristica Universalis*For more about the

**,***universal***, <<**__dialectical__**>> process of***aufheben**'*[*self**-*]*meta**-*<<**>>-***monad**ization**'*, see --As the focus of this blog entry, we wish to elaborate on the point that those two keys, stated above, to overcoming the obstacles to actualization of a <<

**>>, imply a generic conceptual***Characteristica Universalis***of the***completion***of <<**__dialectic__**>> and <<***arithmos***>> which is conceptually***monad***complete in, e.g., Euclid and Aristotle, not to mention in hordes of sub-dialectical thinkers ever since.**__in__The Euclidean/Aristotelian account of <<

**>> and <<***arithmos***>> instantiates yet again the***monad***and***rigidity***-of-conception that characterizes the pre-dialectical stage of human cognitive development, which Plato called <<***fixity***>>, or "the understanding" [in contrast to <<***dianoia***>> or <<***noesis***>>], which Hegel called <<***dialektike'***>>, and which***verstand***F**.**.**__E__**. calls <<**__D__*dia**-*>>.**noesis**Thus --

**"**EUCLID defines in the

*Elements*, VII, 2, a

**as**

*number**“*[i.e., as the <<

**the multitude made up of units**”**>> made up of «**

*arithmos***» -- M.D.], having previously (**

*monads**Elements*VII, 1) said that a

**is "that by which each of existing things is called one". As**

*unit***, neither EUCLID nor ARISTOTLE regard a**

__a____unit____is____not____composed____of____units__**as a**

*unit***, but rather as "the**

*number***of counting, or as the**

*basis***[i.e., as the «**

*origin***» -- M.D.] of**

*archÃ©***."**

*number*[H. Hermes, et al.,

**, Springer-Verlag [NY:**__Numbers__**1991**], page**12**,__emphasis____added__*by**M.D., excerpted from***,**__A Dialectical Theory of Everything__**Preface**].-- so that the Euclidean and Aristotelian modes of thought rigidly '

**dual**ize' the**categories of <<***opposing***>> and <<***arithmos***>>, denying any conceptual "interpenetration" of those***monad***categories.***opposite*The generic conceptual completion of the

**of <<**__dialectic__**>> and <<***arithmos***>> involves the conceptual realization that each [post-<<***monad***>>] <<***arche'***>> of each [post-<<***monad***>>] self-hybrid <<***arche'***>> is also a***arithmos*__sub__*-*<<**>> of the <<***arithmos***>> of its predecessor self-hybrid <<***monads***>>.***arithmos*This realization entails that of the mutual, interconnected

*"***-**__intra__**dual**ities" of the term <<**>> and of the term <<***arithmos***>>.***monad*That is, each [post-<<

**>>] <<***arche'***>> is '***monad***dia**' / '**duo**' /**dual**, 'hiddenly harboring' <<**>> within itself, even as each <<***arithmos***>>, or***arithmos*__sub__*-*<<**>>, can also be grasped as a kind of***arithmos***[y]; a kind of <<***unit***>>, in its own right.***monad*That the

**, the "***unit**unit**-*y" -- that each <<**>> -- is '***monad***dia**'; is*'*-- internally divided -- may even be admitted, at times, informally, by <<**-**__intra__**dual**'**>>, or**__dia__noia*"*, as well as by <<**-**__dia__**noesis**"**>>, or <<***noesis*__dia__*-*>>.**lektike'**But in

*"*, the '**-**__dia__**noesis**"**dia**-' moment of*"*y" is, at best, retained in the form of a rigid conceptual fixation.**unit**-Whereas, on the contrary, in <<

**>>/<<***noesis***>>, the***dialektike'**"***-**__intra__**dual**ity" of*"*y" -- the 'inner**unit**-**two**-ness' of "**one**-ness" -- is grasped as "surpassable", and is surpassed.It is overcome in the form of a new, higher

*"*y", but a new, higher**unit**-*"*y" which has its own new, higher**unit**-*"***-**__intra__**dual**ity" as well; a new, higher*"*y" which is itself also**unit**-*"***-**__intra__**dia**", but in yet a**itatively new, previously unprecedented way.**__qual__Therefore, a true

**-- a true <<***arithmetic***>>, <<***arithmetike'***>>-<<***arithmoi***>>, or***tike'***>>-<<***arithmoi***>> -- must also be a <<***techne'***>>, a***dialektike'***, i.e., it must embrace the**__dialectical__arithmetic__dialectical__**/***opposition**"*/**interpenetration**"**/***interconversion***of the***interconnexion*__dialectical__**<<***opposites***>> and <<***arithmos***>>.***monad*In particular, a true/full

**must encompass / embrace / manifest, and***arithmetic***unify, the***progressively***of the**__dialectic____N__**_**versus__N__**axioms-systems -- denoted '**__Q_____N__**_****#**__N__**' -- the**__Q_____dialectical__**between the***opposition*__N__**_**and the__N____Q___**; between the purely-***arithmetics***itative <<**__quant__**>> of the***arithmoi***atural numbers,**__N__**N**, and the purely-**itative <<**__qual__**>> of the***arithmoi***N****"meta-**__Q__**N**atural meta-numbers".And, such a true

**must reconcile***arithmetic*__N__**_**and__N__**in the form of a --**__Q___**infinite --**__potentially__**categories/axioms-systems "Goedelian"**__dialectical__**of ever new, ever-richer[, e.g., ever richer in descriptive capability], ever-Goedel-***progression***complete, but also ever-more-**__in____PARTIALLY__*-*Goedel-**,**__complete__*"*itative" <<**-**__qual__o__quant__**>>-<<***arithmetike'***'>> systems of***dialektike***, each increasingly expressing/actualizing ever higher***arithmetic**"*ies" of**unit**-__N__**_**and__N__**; of the purely-**__Q___**itative <<**__quant__**>>/***monad***[y] <<***unit***>> number/numeral of***arche'***N**, namely,**1**, with the purely-**itative <<**__qual__**>>/"***monads**unit**-*ies" "meta-numbers"/"meta-numerals" of**N****, namely**__Q____q__**/1**,__q__**/2**,__q__**/3**, ..., in terms of both their**s and their***unit***/***assemblages***/***multiplicities***/***aggregates***/***sums**"*of**cumula**"**s, and such that the [meta-]numeral***unit***s of successor self-hybrid***unit***are -- syntactically as well as semantically --**__dialectical__arithmetics*"*of the**meta**-**unit**-**izations**"*unit**meta**-*of their predecessor self-hybrid**numerals****.**__dialectical__arithmeticsGiven

**n**__=__**1****+...+****1**in**N**, and**m**,**m****n**,**m****z**,**u****n**, and**u****m**, all in/from**N**, the**of the "[meta-]numerals" within the***progression***F**.**.**__E__**.**__D__*systems**-*of**progression**__dialectical__**for the first eight axioms-systems of***arithmetics***can be [crudely] rendered as below, with '**__dialectical__arithmetic**--->**' marking succession, with the "subscript" '__' marking__**o***"*", and with the underscores indicating the**ifiable**__quant__**ifiers**__qual__*"***contra**-**Boolean**"**ifier characteristic [i.e., that --**__qual__

__xx__

**~**

**=**

__x__-- or that --

__xx__

**=**

__x__**+**

__delta_____x__-- which is

**itatively**__qual__**equal to just --**__un__**] --**__x__**n**

**[**

**ified generic**

__unqual__**ifier numerals]**

__quant__**--->**

__q__**/**

**n**

**[**

**ifiable**

__unquant__

*ontological***ifier meta-numerals]**

__qual__**--->**

**u**

**n**

__u__

**o****/**

**n**

**[**

**ifiable**

__quant__

*ontological***ifier meta-numerals]**

__qual__**--->**

__m__**/**

**n**

**[**

**ifiable**

__unquant__

*metrical***ifier meta-numerals]**

__qual__**--->**

**m**

**n**

__m__

**o****/**

**n**

**[**

**ifiable**

__quant__

*metrical***ifier meta-numerals]**

__qual__**--->**

__m__**/(**

__q__**/**

**n**

**..**

**+...+**

**..**

__q__/m)**...**[

**ifiable**

__unquant__*"*able" metrical

**compound****ifier meta-numerals]**

__qual__**...**

**--->**

**m**

**x**

__m__

**o****/(**

**u**

**n**

__u__

**o****/**

**n**

**..**

**+...+**

**..**

**u**

**m**

__u__

**o****/**

**m**

**)**

**...**[

**ifiable**

__quant__*"*able" metrical

**compound****ifier meta-numerals, i.e., fully-algorithmic "dimensional analysis" capability]**

__qual__**...**

**--->**

__a__**...**[

**ifiable [super**

__unquant__**^0**-]

*system***ifier meta-numerals]**

__qual__**...**

**--->**

**...**

**...**

Regards,

Miguel

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