Part 01. 'Beyond the nQ_' Series. The nU_ Dialectical Arithmetics/Algebras.

 

The NU Dialectical Arithmetics/Algebras.

 

Part 01: 'Beyond the NQ' Series.

 

 

 

 

 

Dear Reader,

 

 

 

It is my pleasure, and my honor, as an elected member of the Foundation Encyclopedia Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.

 

This 1st release of this new such series is posted below [Some E.D. standard edits have been applied, in the version presented below, by the editors of the F.E.D. Special Council for the Encyclopedia, to the direct transcript of our co-founder’s discourse].

 

 

 

 

Seldon --

 

The [meta-]systematic-dialectical presentation of the dialectical categorial progression that is the axioms-systems-progression of the F.E.D. 'arithmetics for modeling dialectics', begins with the category of the axioms-system variants of the N_, first-order-logic-only, first-four Peano-Dedekind postulates category, as its <<arche'>>-category of axioms-systems variants.

That progression then moves next to the '''evolute''', external <<aufheben>>-conservation of that N_ category of variant axioms-systems, PLUS its_ internal <<aufheben>>-conservation/-elevation in the revelation of the the N_Q_ category of variant first-order-logic axioms-systems, which constitutes the 'categorial antithesis', i.e., the 'first contra-category', to N_ as the 'categorial first thesis', or <<arche'>>-category.

The qualitative opposition between the N_Q_ systems-category and the N_ systems-category can be grasped as follows.  

The first-order-logic N_ systems are systems of "pure", unqualified arithmetical quantifiers, but without any arithmetical qualifiers to quantify.  

These, the "first-order-logic" Natural Numbers, of the N space internal to axioms-systems category N_, are, principally, 'ordinal quantifiers', representing '''ordinal quantities''', e.g., first, second, third, ... etc.

The NQ space -- internalized into the first-order-logic axioms-systems-variants category N_Q_ -- is made up out of 'unquantifiable ordinal qualifiers', devoid of any arithmetical quantifiers, representing, prior to any further layers of their interpretation, the 'ordinal qualities' succession of 'first-NESS', 'second-NESS', 'third-NESS', etc.

This opposition is thus a matter of "pure" arithmetical quantifiers versus "pure" arithmetical qualifiers.

This dialectical, categorial progression, systems-progression, 'meta^1-system', or 'meta-systematic dialectical' presentation, of the F.E.D. 'arithmetics for modeling dialectics', is made up out of a heterogeneous multiplicity of [mere-]systems [meta^0-systems]. 

It next moves on, from the apparent antithesis, and, even, apparently, the "radical dualism" or "Kantian Antinomy", of N_ versus N_Q_, to the resolution of this opposition, in a third category of dialectical-arithmetical/algebraic axioms-systems variants.

That third axioms-systems-category forms a 'categorial synthesis', a '''complex unity''', or a 'uni-category' with respect to the categories N_ and N_Q_ systems.  

We call this category N_U_, which incorporates the 'meta-number' "space" NU.  

This dialectical synthesis systems-category can also be signified by q_QN. 

The 'dialectical meta-numbers' that populate the NU space are of the form -- 

u(t)n x u^on 

-- within which n is a[ny] "Natural" Number, u(t)n is the quantifier for the units of the nth ontological category of the multi-category Domain being modeled, and within which 'x' denotes the NU multiplication operation, and u^on is the quantifiable [^o] unit-qualifier for the nth ontological category of the Domain being modeled.  

A qualitative unit of a given ontological-categories Domain, denoted u^on, represents the individual [ev]entities, all of kind n, that are collectively and univocally represented by the nth ontological category of the Domain being modeled.

Thus, the 'meta-numbers' of the form u(t)n x u^on state the mean population count of units of the nth ordinal kind category, as of/during historical epoch t, for the multi-kind, multi-population Domain being modeled using the NU arithmetic/algebra.

Thus, the 'quantifiable qualifiers', or, equivalently, the qualifiable quantifiers' -- the space NU 'meta-numbers' -- combine and unify the 'unqualified quantifiers' of space N -- e.g., 

N = {1, 2, 3, ...} 

-- with the 'unquantifiable qualifier meta-numbers' of space NQ -- e.g., 

NQ = {q1, q2, q3, ...} 

-- in a category of the first 'dialectical synthesis axioms-systems' of the F.E.D. 'arithmetics for modeling dialectics', whose space is --

NU = {{u(t)1 x u^o1}, {u(t)2 x u^o2}, {u(t)3 x u^o3}, ...}.   

 

 

 

 

 

 

 

 

For more information regarding these Seldonian insights, please see --

 

www.dialectics.info

 

 

 

 

 

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ – as well as dialectically-illustrated books published by the F.E.D. Press, see –

 

https://www.etsy.com/shop/DialecticsMATH

 

 

 

 

 

 

 

¡ENJOY!

 

 

 

 

 

 

 

Regards,

 

 

Miguel Detonacciones,

 

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

 

 

 

 

 

 

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