Part 11: Seldon’s Axioms Series. An Apparent Anomaly of the Dialectic-Arithmetics’ Dialectic that does not Afflict the, parallel, Algebraic-Logics’ Dialectic.

 

 

 

Part 11: Seldon’s Axioms Series.

 

 

An Apparent Anomaly of

the Dialectic-Arithmetics’ Dialectic

that does not Afflict the, parallel,

Algebraic-Logics’ Dialectic.

 

 

 

 

 

 

 

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 elements of Seldonian Theory.

 

The 11th text in this new such series is posted both above and 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 “standard”, “first-order logic” W-arithmetic axioms-system, whose four ‘Peanic’ axioms emphasize the “ordinal number”, not the “cardinal number”, aspects of the “Whole numbers” space, W = {0, 1, 2, …}, work for the first triad of the ‘meta-systematic dialectic’ of the unit-interval-confined arithmetics for modeling algebraic Logics – for their Boolean formal-Logic-arithmetics «arché»-category, _{W_}E__L, for its first 

‘contra-category’, for modeling dialectical Logics,

_{W_}Q__L, and for their first ‘uni-category’,

_{W_}q _QE^L, only because the only values from W that all 

three of these ‘ideo-ontological’ categories involve are limited to the unit-interval of W – indeed, only to just its endpoints, to just {0, 1}.  Thus, the latent cardinality of the W values does not obtrude.”

 

“For example, in the case of the _{W_}q _QE^L category of 

arithmetic-systems for modeling Logics – for modeling ‘actualization/extinction Logics’ in its case – the quantifying ‘‘‘coefficients’’’ of its quantifiable [‘º’] class or categorial arithmetical qualifiers, of the form b_k(t) in b_k(t)uº_k, generically, are such that the value b_k(t) is either 1, if the kth class or ontological category is extant in stage t, or 0 if that ontic class is absent in, e.g., is extinct by, stage t.

 

“However, in the case of the ‘meta-systematic dialectic’ of the ‘‘‘full-multiplicity arithmetics’’’ for modeling dialectics full-blown, and not just the unit-interval-confined arithmetics for the algebraic logics, i.e., for full-blown dialectics Domain ^_ 

D = #_

^_ with W_^#   as the «arché»-category this time, again subsuming only the “first-order logic”, ‘Peanic’, ordinal-number-emphasizing axioms for the “standard” W_ axioms-system, the corresponding third axioms-systems category, the ‘first uni-category’ of this dialectic, denoted by _{W_}q _QW^#_, encounters an apparent anomaly.”

 

“The generic ‘meta-numeral’, the ‘quantified [‘º’] 

qualifier’, of the _{W_}q _QW^#_ axioms-systems category of 

dialectical arithmetics is u_k(t)uº_k, such that the 

quantifiable [‘º’] arithmetical-qualifier, uº_k, stands for 

the generic «monad» or unit of the kth ontological category, or «arithmos».  

Its full-multiplicity arithmetical quantifier, u_k(t), stands for a number in W_, that represents the 

average population count of “individual units” of the kth kind, inhering in ontological category k during stage or period t.”

 

“It thus appears that u_k(t) must stand for a “cardinal” 

– or “counting” – number.” 

 

“But the numbers contained in the set/space W  [double-underscore], i.e., for its first-order-logic-only, ‘Peanic’ axioms [double-underscore], are not, explicitly, the “cardinal numbers” that are emphasized by the first-and-second-order-logic axioms for the “standard Whole numbers”, denoted herein by W [single-underscore].”

“The W_ numbers subsumed in _{W_}q _QW^#  are, explicitly, only “ordinal numbers”.”

 

“Where, then, do the population count, full-multiplicity arithmetical quantifiers, the {u_k(t)}, come from, if their arithmetical source for modeling full-blown dialectics, is W_ [“ordinal” whole numbers] and not W [“cardinal” whole numbers]; is –

_{W_} q _QW^#_

– and not –

 

_Wq _QW^#_?” 

 “This apparent anomaly dissolves however – disappears – if we consider that the ‘“population”’ of units that “make up” a given “kind” or ontological category, at a given stage or period, t, of that kind’s history, should themselves be ‘ordinalized’.”

 

“This could be done by assigning to each individual unit a strictly-consecutive, different ordinal number, starting with/from 1 for the ordinal whole number “first”, either per some systematic principle of those units’ ordering, e.g., in terms of their “individual differences”, or by “random” assignment of the needed ordinal numbers to ordinally-assign each and every unit of the entire kind-k ‘‘‘population’’’.” 

 

“Then, the value of a given u_k(t) in a given term u_k(t)uº_k in a ‘qualo-quantitative’ series, or “non-amalgamative sum” [cf. Musès], of such terms, for the progression of ontological categories, or kinds, that is relevant for period t of the given Domain, would be the highest ordinal number assigned to the individual units or «monads» of the kth kind, the kth «arithmos», the kth ontological category, for, e.g., the average ‘‘‘population’’’ count of category k units during period t.”

 

 

 

 

 

 

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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