Monday, October 24, 2011

The Dialectic of "Arithmos" and "Monad"

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




F.E.D. holds that the keys to the practicable actualization of such a <<Characteristica Universalis>> are:

(1) The realization of the [ideo- and physio-]ontological fecundity of the 'self-combinations' of the <<arche'>> ontological category, or term, of a categorial progression, or terms-progression -- as opposed to presuming the ontological sterility of such 'self-combinations' [a la Leibniz's and Boole's ontologically-reductionist and Parmenidean "fundamental law of thought", xx = x, postulated by them for every ontological category/term x], and;

(2) Interpretation of such fecund 'self-combinations' in terms of the '[self-]meta-<<monad>>-ization' of the <<monads>> of each predecessor self-hybrid <<arithmos>>-of-<<monads>>, as the name/description of the process whereby the <<monads>> of its successor self-hybrid <<arithmos>>-of-<<monads>> -- in such categories/<<arithmoi>> progressions -- are constructed.

These keys to the F.E.D. breakthrough to a dialectical <<Characteristica Universalis>> are slated to be detailed in a future entry to this blog. 

For more about the universal, dialectical, <<aufheben>> process of '[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 <<Characteristica Universalis>>, imply a generic conceptual completion of the dialectic of <<arithmos>> and <<monad>> which is conceptually incomplete in, e.g., Euclid and Aristotle, not to mention in hordes of sub-dialectical thinkers ever since.




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

Thus --

"EUCLID defines in the Elements, VII, 2, a number as the multitude made up of units [i.e., as the <<arithmos>> made up of «monads» -- M.D.], having previously (Elements VII, 1) said that a unit is "that by which each of existing things is called one".  As a unit is not composed of units, neither EUCLID nor ARISTOTLE regard a unit as a number, but rather as "the basis of counting, or as the origin [i.e., as the «arché» -- M.D.] of number."

[H. Hermes, et al., Numbers, Springer-Verlag [NY: 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 'dualize' the opposing categories of <<arithmos>> and <<monad>>, denying any conceptual "interpenetration" of those opposite categories.


The generic conceptual completion of the dialectic of <<arithmos>> and <<monad>> involves the conceptual realization that each [post-<<arche'>>] <<monad>> of each [post-<<arche'>>] self-hybrid <<arithmos>> is also a sub-<<arithmos>> of the <<monads>> of its predecessor self-hybrid <<arithmos>>.

This realization entails that of the mutual, interconnected "intra-dualities" of the term <<arithmos>> and of the term <<monad>>.

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

That the unit, the "unit-y" -- that each <<monad>> -- is 'dia'; is 'intra-dual' -- internally divided -- may even be admitted, at times, informally, by <<dianoia>>, or "dia-noesis", as well as by <<noesis>>, or <<dia-lektike'>>.

But in "dia-noesis", the 'dia-' moment of "unit-y" is, at best, retained in the form of a rigid conceptual fixation.

Whereas, on the contrary, in <<noesis>>/<<dialektike'>>, the "intra-duality" of "unit-y" -- the 'inner two-ness' of "one-ness" --   is grasped as "surpassable", and is surpassed. 

It is overcome in the form of a new, higher "unit-y", but a new, higher "unit-y" which has its own new, higher "intra-duality" as well; a new, higher "unit-y" which is itself also "intra-dia", but in yet a qualitatively new, previously unprecedented way.


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

In particular, a true/full arithmetic must encompass / embrace / manifest, and progressively unify, the dialectic of the N_ versus NQ_ axioms-systems -- denoted 'N_ # NQ_' -- the dialectical opposition between the N_ and the NQ_ arithmetics; between the purely-quantitative <<arithmoi>> of the Natural numbers, N, and the purely-qualitative <<arithmoi>> of the NQ "meta-Natural meta-numbers".

And, such a true arithmetic must reconcile N_ and NQ_  in the form of a -- potentially infinite -- dialectical categories/axioms-systems "Goedelian" progression of ever new, ever-richer[, e.g., ever richer in descriptive capability], ever-Goedel-incomplete, but also ever-more-PARTIALLY-Goedel-complete, "qualo-quantitative" <<arithmetike'>>-<<dialektike'>> systems of arithmetic, each increasingly expressing/actualizing ever higher "unit-ies" of N_ and NQ_; of the purely-quantitative <<monad>>/unit[y] <<arche'>> number/numeral of N, namely, 1, with the purely-qualitative <<monads>>/"unit-ies" "meta-numbers"/"meta-numerals" of NQ, namely q/1, q/2, q/3, ..., in terms of both their units and their assemblages / multiplicities / aggregates / sums / "cumula" of units, and such that the [meta-]numeral units of successor self-hybrid dialectical arithmetics are -- syntactically as well as semantically -- "meta-unit-izations" of the unit meta-numerals of their predecessor self-hybrid dialectical arithmetics.

Given n  =  1+...+1 in N, and m, mn, mz, un, and um, all in/from N, the progression of the "[meta-]numerals" within the F.E.D. systems-progression of dialectical arithmetics for the first eight axioms-systems of dialectical arithmetic can be [crudely] rendered as below, with '--->' marking succession, with the "subscript" 'o' marking "quantifiable qualifiers", and with the underscores indicating the "contra-Boolean" qualifier characteristic [i.e., that --
xx ~= x 

-- or that --

xx  =  x + delta_x

-- which is qualitatively unequal to just -- x ] --

n [unqualified generic quantifier numerals] ---> 

q/n [unquantifiable ontological qualifier meta-numerals] --->

unuo/n [quantifiable ontological qualifier meta-numerals] --->

m/n [unquantifiable metrical qualifier meta-numerals] --->

mnmo/n [quantifiable metrical qualifier meta-numerals] --->

m/(q/n..+...+..q/m)...[unquantifiable "compoundable" metrical qualifier meta-numerals]...--->

mxmo/(unuo/n..+...+..umuo/m)...[quantifiable "compoundable" metrical qualifier meta-numerals, i.e., fully-algorithmic "dimensional analysis" capability]...--->

a...[unquantifiable [super^0-]system qualifier meta-numerals]...

--->......






Regards,

Miguel
 

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