Part 11: Seldon on ‘‘Dialectical Categorial Progressions’’’ Series. Dialectical Ontological Multiplication Axioms – Theme & Variations.

 

 

 

Part 11: Seldon on

 

‘‘‘Dialectical Categorial Progressions’’’ Series.

 

 

Dialectical

 Ontological

Multiplication

Axioms –

 

Theme

&

Variations.

 

 

 

 

 

 

 

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, Seldon’s commentaries on key Encyclopedia Dialectica concepts of Seldonian Theory.

 

 

This 11th text in this series is posted herewith, together with supporting text-images and diagrams [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 dyadic, triadic and tetradic dialectical functions provide our most compact means of expressing the self-iterative and self-involutionary processes which generate dialectical, ontological-categorial progressions.”

 

“These dialectical functions – these «aufheben» functions – do not, however, provide the most expeditious ways to actually calculate the categorial progressions whose «aufheben» principle of gene-ration these functions model.”

 

“For that, we recommend the ‘Qualo-Peanic Method’.  That method-name means to generate each consecutive next category-symbol, to be summed-in with the other category-symbols already generated, starting with the «arché» category-symbol, by subscript-appending that «arché» category-symbol, as the new final subscript, to each previously-generated, previously most-advanced category-symbol.  This procedure is the ‘qualitative equivalent’ of the Peano successor function; adding 1 to each previously-meristemal “Peano Natural Number”, n, to form its successor: n ---> s(n) = n+1.” 

 

“Thus, if q_a is the «arché» category-symbol for a generic dialectical, ontological-categorial progression into increasing categorial complexity or “determinateness”, then this method is to generate the second category-symbol as q_aa.  If you can solve that category as q_b, wherein ‘b’ abbreviates a univocal name for the quality that results from the «aufheben» ‘self-meta-unit-ization’ of the units of category/quality q_a, then your categorial progression now consists of two non-amalgamatively-summed (‘[+]’) and solved [defined] terms/category-symbols – 

q_a [+] q_b.”

 

“Next, append subscript a again, this time to new ‘meristemal’ or “vanguard” category-symbol q_b, yielding q_a [+] q_b [+] q_ba, wherein q_ba hints at a dialectical-categorial synthesis of category-defining quality b and/with category-defining quality a.  If you can solve q_ba with a univocal quality-name, kind-name or category-name, ‘abbreviate-able’ by c, then you now have a categorial progression with three category-symbol terms: 

q_a [+] q_b [+] q_c.  

And so on… .”

 

“The key to this ‘Qualo-Peanic Method’ is to maintain strict consistency in your solutions to ‘multi-vocal’ terms [i.e., initially, algebraic-unknown terms], e.g., to pairs of identical subscripts, solving each such case with the same, exact determination-symbolizing subscript-symbol with which you replaced that pair of subscripts previously, i.e., with a single subscript specific to that pair, each time that specific subscript pair, or ‘multi-vocality’, [re-]occurs.” 

 

“Nevertheless, the dialectical functions, which are all «aufheben» functions – which all represent «aufheben» operations – are the keys to compactness in the representation of dialectical, onto-logical-categorial, progressions, and it is valuable to know the rules of their calculation, even if you don’t actually usually calculate the categorial progression thereby.” 

 

“The central rule for such a calculation is a ‘dialectical ontological multiplication rule’, or “product-rule” [defining ‘[´]’], which, in our standard renditions of the core axioms for our ‘first arithmetic/algebra for modeling dialectics’, we present as ‘Axiom §9.”

 

“While we have not exhaustively explored all possible ‘product rule axioms’ for this ‘first dialectical arithmetic’, we have explored four variants.  The ‘gene-ral’ ‘‘‘theme’’’ of all four variants is the «aufheben», or dialectical, process, but each variant represents a ‘‘‘variation’’’ on that ‘‘‘theme’’’, focusing on a different possible definition of a different “species” of the «aufheben» process.”

 

“The first variant, that we most often use and cite in our publications is:

 

Axiom §9.  ‘The double-conservation «aufheben» evolute product’:

 

For all j and k in N, 

q_j [´]  q_{k  =}  q_k [+]  q_j+k.”

 

“Per it, the operand or multiplicand is «aufheben»-conserved in the resulting non-amalgamative sum, but not the multiplier.  Note also that, if j = k, then 

q_j [´]  q_{j  =}  q_j [+]  q_{j+j  =}  q_j [+]  q_2j.  

If we re-express the generic arithmetic’s rule using generic category subscripts, we get –

q_B [´]  q_{A  =}  q_A [+]  q_BA 

and

q_A [´]  q_{A  =}  q_A [+]  q_AA 

 

– wherein we see that, in the first example, q_Ais doubly conserved, both internally and externally.  That is, q_A is conserved inside q_BA, and also outside of q_BA, as the external summand q_A.  Hence the terms “double-conservation” and “evolute” in the phrase that names/describes this product-rule axiom.”

 

“The second product-rule axiom variant that we have explored is in a way the opposite of the first:

 

Axiom §9.  ‘The meta-catalytic «aufheben» evolute product’:

 

For all j and k in N, 

q_j [´]  q_{k  =}  q_j [+]  q_j+k.”

 

“Per it, the operator or multiplier is «aufheben»-conserved in the resulting non-amalgamative sum, but not the multiplicand or operand.  Note also that, if j = k, then 

q_j [´]  q_{j  =}  q_j [+]  q_{j+j  =}  q_j [+]  q_2j.  

If we re-express the generic arithmetic’s rule using generic category subscripts, we get –

 

q_B [´]  q_{A  =}  q_B [+]  q_BA 

 

and

 

q_A [´]  q_{A  =}  q_A [+]  q_AA 

 

– wherein we see that, in the first example, q_Bis doubly conserved, both internally and externally.  That is, q_B is conserved inside q_BA, and also outside of q_BA, as its external summand q_B.”  

“Thus we see that the action of q_B is somewhat analogous to that of a catalyst in a chemical reaction.”

 

“The third product-rule axiom variant that we have explored is in a way the combination or “synthesis” of the first two:

 

Axiom §9.  ‘The meta-genealogical «aufheben» evolute product’:

 

For all j and k in N, 

q_j [´] q_{k  =}  q_j  [+]  q_k [+]  q_j+k.”

 

“Per it, the operator or multiplier and the operand or multiplicand are «aufheben»-double-conserved in the resulting non-amalgamative sum.  Note also that, if j = k, then –

q_j [´] q_{j  =}  q_j [+] q_j [+] q_{j+j  =}  q_j [+] q_2j 

 – with the redundant q_j dropping out per our ‘Axiom §7’, the ‘unquantifiability axiom’: 

q_k [+]   q_{k  =}  q_k.”   

 

“If we re-express the generic arithmetic’s rule using generic category subscripts, we get –

q_B [´]  q_{A  =}  q_B [+] q_A [+] q_BA 

and

q_A [´]  q_A =  q_A [+]  q_A [+]  q_AA =  q_A [+]  q_AA 

 

– wherein we see that, in the first example, both q_Band q_A are doubly conserved, both internally and externally.  That is, q_B is conserved inside q_BA, and also outside of q_BA, as the external summand q_B, and also q_A is conserved inside q_BA, and also outside of q_BA, as the external summand q_A.” 

“Both “parents”, q_A and q_B, continue to exist after their multiplicative interaction, together their “child”, the q_BA net result of that multiplicative interaction.  Hence the terms ‘meta-genealogical’ and “evolute” in the phrase that names/describes this product-rule axiom variant.  This third product-rule variant is also commutative, unlike the first two variants.”

 

“Note: All three of the ‘evolute’ variants of Axiom §9. produce the same categorial-progression results, via the dyadic and triadic dialectical functions.  

 

“The fourth variant is a non-‘evolute’ variant, in which neither the «aufheben»-multiplier nor the «aufheben»-multiplicand is conserved externally.  Only internal conservation is recognized in this ‘convolute’ product rule:

 

Axiom §9.  ‘The meta-heterosis «aufheben» convolute product’:

 

For all j and k in N, q_j [´] q_{k  =}  q_j+k 

[although this case cannot arise for any of our standard dialectical functions, dyadic, triadic, and tetradic alike].”

 

“Per it, the operator or multiplier and the operand or multiplicand both disappear into their categorial-dialectical synthesis-category or antithesis-category, ‘uni-category’, q_j+k, or ‘contra-category’ q_j+j.”  

“Note that, if j = k, then –

q_j [´] q_{j  =}  q_{j+j  =}   q_2j.”   

 

“If we re-express the generic arithmetic’s rule using generic category subscripts, we get –

q_B [´]  q_{A  =}  q_BA 

and

q_A [´]  q_{A  =}  q_AA 

 

– wherein we see that, as a result of this version of Axiom §9, the categorial “progression” is no longer a non-amalgamative series or sum of a growing number of category-symbols standing for a growing number of qualitatively-different, ontologically-different, different-in-kind ontological categories.  Instead, via our standard dialectical functions, it is a parade of symbolic successor categorial antithesis after predecessor categorial antithesis, with each symbolic predecessor antithesis-category superseded and replaced by its symbolic successor antithesis-category, after each ‘auto-«aufheben»’ self-operation [“squaring”, for the dyadic function] of predecessor antithesis-category-symbol.”

“If we exemplify this via categories of the, dyadic, ‘Dialectic of Nature meta-model’, we get  q_x^2t, with the, Peanic, successorship rise in value of whole number ‘meta-power’ t, as –

q_x² = q_c; 

q_c² = q_r; 

q_r² = q_a; 

q_a² = q_m; 

q_m² = q_p; 

q_p² = q_e;…  

 

– which models a cosmos in which every prior kind converts completely into its successor kind, without anything of the old continued into the new.”

 

“We recognize that this kind of, ‘convolute’, «aufheben» process is what is implicit in most descriptions of ‘dialectical, «aufheben» determinate negations’, in the history of dialectics, outside of the writings of Hegel, and, e.g., of the implicit categorial dialectics of the table of contents of Marx’s Capital, volume I.”

 

“However, this ‘convolute’ variant of Axiom §9 fails to fit the empirical observations of dialectic in exo-human/pre-human Nature, and in human Nature, alike.”

 

 

 

 

 

 

 

For more information regarding these Seldonian insights, and to read and/or download, free of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:

www.dialectics.info

 and

https://independent.academia.edu/KarlSeldon

 

 

 

 

 

 

 

 

 

 

 

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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