Part 07: Seldon's Series on ‘‘‘Dialectical Categorial Progressions’’’. CARDINAL FEATURES of Their ‘Content-Structure’.

 

                 

 

 

 

 

Part 07: Seldon's Series on

 

 

‘‘‘Dialectical Categorial Progressions’’’.

 

 

CARDINAL

FEATURES

 

of

 

Their

‘Content-

  Structure’.

 

 

 

 

 

 

 

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 7th text in this new such 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 special ‘dialectogram’ diagram, posted above, atop the other images also posted above, is what I want to use to address some salient and cardinal aspects of the ‘content-structure’ of dialectical categorial progressions, especially those generated using the Encyclopedia Dialectica ‘dyadic dialectical function’.”

 

“The categorial progression example illustrated in that special ‘dialectogram’ is that for our ‘dialectic of nature meta-model meta-equation’, starting from a ‘‘‘Dark Energy’’’ – from our ‘Regenerist’ hypothesized ‘spandetron’ – «arché»-category; for our, highest-content, known ‘cosmo-ontological’ categories’ ‘Dialectical Theory of Everything’ master equation, which can stand in for myriad other dialectical categorial progression ‘meta-models’ for the myriad dialectics going on ‘inside’ these high-level categories, at ‘qualo-fractal’ scales scaled-down from that highest scale.”

 

“Note: The last three, bottom-most category-symbols in the ‘dialectogram’ at top, represent our predictions, or ‘symbolic ontological pre-constructions’, of the future of the dialectic of nature, at least in our planetary vicinity.  Even the category denoted h, for ‘humanitys’, stands for, in particular, planetary ‘humanitys’ that have achieved their, predicted by us, higher destinies, each unit a ‘political-ECONOMIC-DEMOCRATIC’ ‘planetary polis’, made up out of a heterogeneous multiplicity of globally-federated sovereign nation-state units as its sub-units.”

 

“Consequently, the next ‘self-hybrid’ category, denoted by y in that ‘dialectogram’, stands for the category for ‘meta-humanity’ units, each such unit being an «aufheben» ‘meta-unit-ization’ of two or more such ‘planetary polis’ units, forming, as its ‘meta-units’, an, also ‘political-ECONOMIC-DEMOCRATIC’, interplanetary federation, but still within a single stellar/planetary system, each one made up out of a heterogeneous multiplicity of ‘planetary polis’ units as its sub-units.”

 

“Again, consequently, the still-next ‘self-hybrid’ category in this, cosmological, dialectic of nature categorial progression – the category denoted by z in that ‘dialectogram’ – stands for an «arithmos» of ‘meta-meta-humanity’ units, or ‘meta²-humanity’ units, each such unit being an «aufheben» ‘meta-unit-ization’ of two or more such inter-planetary federation units, each within a distinct, single stellar/planetary system, forming, as ‘meta-units’, an, also ‘political-ECONOMIC-DEMOCRATIC’, multiple stellar/-planetary systems interplanetary federation, i.e., an inter-stellar/planetary systems unit, each one made up out of a heterogeneous multiplicity of single stellar/planetary system interplanetary federation units.” 

 

“One can, of course, imagine beyond category z, envisioning galaxy-wide meta³-human social formations, and on to inter-galactic social formations – such as might even possibly already exist elsewhere in our universe.  But let us, for present purposes, confine ourselves to the ‘Star Trek Hypothesis’, or to the ‘Star Wars Hypothesis’, or to the ‘Star Gate Hypothesis’, and stop our ‘pre-constructions’ at the – already fabulously ambitious – multiple stellar/planetary systems interplanetary federations ‘qualo-fractal’ scale of meta-human social formation.  The ‘social self-force of human-societal expanded self-re-production’ is, at present, too limited to give such further extrapolations of this universal categorial progression much ‘thought-concreteness’.”

 

“Now, let’s get to noticing the cardinal ‘content-structure’ features of this, ultimate, dialectical categorial progression, that might be general features, applicable to the whole category of ‘dyadic dialectical function’-generated dialectical categorial progressions.”

 

“First off, please note that the ‘“epochs”’ being defined here, so as to showcase the cardinal, quantitative features of _NQ, ‘dyadic- dialectical-function-generated’ progressions, series or [non-amalgamating] ‘‘‘sums’’’ of heterogeneous, qualitatively distinct category-symbols, are not the standard ‘“epochs’’’ of which we usually speak and write, in this, categorial progressions’ context.  The epochs addressed herein are ‘real subsumption [cf. Marx] epochs, or ‘rs-epochs’ for short.”

 

“The ‘real subsumption epochs’ differ significantly from our standard concept of ‘dyadic dialectical function’, dialectical categorial progression epochality.  Our standard such epochs, after the zeroth epoch, each end with the latest new ‘self-hybrid’ category-symbol, or latest ‘contra-category’ symbol; they end with the latest new ‘antithesis category’ symbol.  They thus represent the ‘“formal subsumption’’’ [cf. Marx] of all priorly-arisen category-symbols by that latest category-symbol, but not their ‘“real subsumption’’’.  The sub-progression of the real subsumptions of all of the previously-posited category-symbols by that latest ‘self-hybrid’ category-symbol are all allocated to the consecutive next standard epoch.”

 

“On the contrary, the ‘real subsumption epochs’ – again, after their zeroth rs-epoch – all begin with that latest, ‘self-hybrid’, ‘contra-category’, ‘antithesis category’ symbol.  They then continue with all of the previously-posited category symbols that have now, all thus being now more-advanced ‘allo-hybrid’ category-symbols, in the given, new rs-epoch, achieved partial dialectical synthesis with – or, finally, full dialectical synthesis with – that latest ‘antithesis category-symbol’, that has thus subsumed, or coalesced with, all of those previously-posited category-symbols.  The given rs-epoch’s categorial sub-progression ends with that epoch’s maximal, and ‘‘‘culminant’’’ ‘allo-hybrid’ category-symbol, whose subscripts include the subscript for that latest ‘antithesis category-symbol’, together with all of the earlier-arisen, category-describing subscripts.” 

 

“That ‘‘‘culminant’’’ ‘allo-hybrid’ category-symbol is the ‘full-synthesis category-symbol’, the fullest dialectical synthesis category-symbol that is possible for the given rs-epoch, and for that epoch’s beginning ‘antithesis category-symbol’; the highest-in complexity-scale category-symbol in the ‘qualo-fractal’ scales-progression of that series of rs-epochs; the topmost of the ‘qualo-fractal’ levels of that category-symbols’ series up to that rs-epoch.

 

“In the example at hand, our first three standard epochs are –

á x ñ à á x <+> c ñ à á x <+> c <+> q_cx <+> r ñ.

– whereas, for rs-epochs, those three are –

á x ñ à á x <+> c <+> q_cx  ñ à

á x <+> c <+> q_cx <+> r <+> q_rx <+> q_rc <+> q_rcxñ.”

 

“Please observe, via the images posted at the top of this blog-entry, that –

1.  Consecutive ordinal powers of the dyad, 2 – starting with/from the zeroth power, 2⁰, as embedded in the function (2^(t+1)) - 1, applied to the first rs-epoch, t = 0 – tell you the total count of distinct, qualitatively-different ‘cosmo-ontological’ category-symbols in the rs-epoch representation generated by that power-of-two function, when the «arché»-category symbol, herein x, for the ‘cosmo-ontological category’ of/for our hypothetical “Dark Matter”/‘spandetron’ units, is raised to that power of two, i.e., ‘self-involutes’ to that, (2^(t+1)) - 1, degree;

 

2.  Each “rs-epoch” begins with a new, unprecedented ‘self-hybrid’ category symbol, and then continues with a non-amalgamating sum of also new, unprecedented ‘allo-hybrid’ category-symbols.  Each of the latter has the subscript representing that new ‘self-hybrid’ category-symbol explicitly constituting the leading, left-most epithet of its multi-epithet, or hybrid, subscript, thereby asserting that the united meaning of all of its right-more subscripted epithets has been subsumed by the meaning of that left-most, new, latest subscripted epithet;

 

3.  For rs-epoch of ordinal number t = n, there are (2^(t+1)) - 1 qualitatively-distinct ‘cosmo-ontological’ category-symbols in that rs-epoch’s categorial progression as a whole.  All of those distinct category-symbols are syntactically and combinatorically possible.  Each epochal category-symbols’ non-amalgamating ‘‘‘sum’’’ marks a ‘possibility space’, with each category occupying a different dimension of that space, so that the ‘dyadic function’, as t increases, forms an ‘‘‘analytical-geometric’’’ space that is [self-]expanding in dimensionality with every successive squaring of that category-symbols-sum, as each t succeeds/exceeds itself to its t + 1.  But that mere syntactic and combinatoric ‘possibilisticity’ does not mean that all of these category-symbols will ever be actual, instantiated, viable, semantically meaningful, in the Domain whose ontological-categorial realities the categorial progression is intended to model.  Some category-symbols may represent null categories, always-empty categories, for the given Domain.  Others, “populated” in one+ rs-epoch(s), may later go extinct, from within a later epoch, & remaining extinct thereafter;

 

4.  For each rs-epoch ordinal number, n, denoted generically by this ordinal variable within N, there will be 2ⁿ – 1 brand new ‘allo-hybrid’ category-symbols, summed together with their one new ‘self-hybrid’, leading category-symbol, in rs-epoch t = n;

 

5.  All of the category-symbols, from epoch t = n = 1 on, are generated by the self-iterated ‘‘‘self-involutions’’’ of the «arché»-category-symbol, x, or q_x, i.e., via, in this example, x^2t, as t = n increases, monotonically and consecutively, recurrently, from each t to its t + 1.”

 

 

 

 

 

 

 

 

 

 

 

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.

 

 

 

 

 

 

YOU are invited to post your comments on this blog-entry below!

 

 

 

 

 

 

  

Part 06: Seldon on ‘Dialectical Categorial Progressions’ Series. How nQ Axioms §7 & §9 “Conspire” Against Redundancy & Unwanted Quantifiability.

  

                       
                         

Part 06: Seldon on

‘Dialectical Categorial Progressions’ Series.

 

 

 How

_NQ Axioms

§7 & §9

 

“Conspire”

Against

Redundancy

&

Unwanted

Quantifiability.

 

 

 

 

 

 

 

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 6th text in this new such 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 –

 

“Tonight I want to show you how the interplay of axioms §7 & §9 of the _NQ “Non-Standard Model” of the Peano-Dedekind “Natural” Numbers – the set, or space, 

N = {1, 2, 3, …} 

– in the computation and formation of dialectical ontological categorial progressions in that _NQ language, blog both potential redundancies and potential ‘lack-cunae’ [E.D. editors: “lacunae”] in the ‘consecuum’ of those progressions, interpreted for a specific Domain of knowledge, and how they thereby – by blocking redundancies – preclude any, unwanted, quantifiability in that, “purely”-qualitative, “purely”-ontological-categorial, _NQ dialectical language.”

 

“If we take, for example, the categorial progression for our current model of ‘the dialectic of nature’ – for the specific Domain that is also the most general, most universal Domain of which we know, D = " – and expand it, using the ‘Dyadic Dialectical Function’, dyadically, up to its epoch _"t₂ = 3, we obtain the following ontic, categorial progression in its raw, axiomatically-unpolished form –

 

Epoch _"t₂ = 0:  q_x²⁰  =  q_x¹  =   q_x;

Epoch _"t₂ = 1:  q_x²¹  =  (q_x²⁰)²  =  q_x²  =  

(q_x) Ä (q_x) =

 

q_x Å q_xx  |-=  q_x ~Å~ q_c; 

Epoch _"t₂ = 2:  q_x²²  =  (q_x²¹)²  =  q_x⁴  = 

(q_x ~Å~ q_c) Ä (q_x ~Å~ q_c) = 

(q_x Å q_xx) Å (q_c Å q_xc) Å 

(q_x Å q_cx) Å (q_c Å q_cc)  |-= 

q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r; 

But: we had two of category-symbol q_x and two of category-symbol q_c.  

How did we get rid of the redundant q_x and q_c?

How did we avoid incurring a ‘quantifiability’, unwanted in the, “purely”-qualitative, _NQ arithmetic – 

how did we avoid ‘2q_x’and ‘2q_c’?”

“Axiom §7, the “additive idempotency” axiom:

 

Axiom §7: For every n in N and for every q_n in _NQ, 

q_n Å q_n  =  q_n.  

So, \, q_x Å q_x  =  q_x, and q_c Å q_c  =  q_c.”  

“Note also that, given the Axiom §9 ‘double conservation aufheben evolute product rule’, we didn’t develop any breaks in the _NQ ‘consecuum’ undergirding this ontic categorial progression either: 

Axiom §9: For every j, k in N 

and for every q_j and q_k in _NQ, 

q_k Ä q_j  =  q_j Å q_k+j.  

Thus, we didn’t get –

(q_x ~Å~ q_c) Ä (q_x ~Å~ q_c) = 

(q_xx) Å (q_xc) Å 

(q_cx) Å (q_cc)  |-= 

q_c ~Å~ q_cx ~Å~ q_r

– leaving out q_x, the “ever-present” arché-category, and thus replacing the ‘evoluteness’ of the aufheben product rule with a, partial, ‘convoluteness’.”  

“[Note: subscripts are ordinally commutative, so that we use q_cx to stand also for q_xc; q_xc to stand also for q_cx].”

 

“Likewise, for –

Epoch _"t₂ = 3:  q_x²³  =  (q_x²²)²  =  q_x⁸  = 

(q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r) Ä

(q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r) = 

(q_x Å q_xx)Å(q_c Å q_xc) Å 

(q_cx Å q_xcx)Å(q_r Å q_xr) Å  

(q_x Å q_cx)Å(q_c Å q_cc) Å 

(q_cx Å q_ccx)Å(q_r Å q_cr) Å  

(q_x Å q_cxx)Å(q_c Å q_cxc) Å 

(q_cx Å q_cxcx)Å(q_r Å q_cxr) Å  

(q_x Å q_rx)Å(q_c Å q_rc) Å 

(q_cx Å q_rcx)Å(q_r Å q_rr)  |-= 

q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r ~Å~        

q_rx ~Å~ q_rc ~Å~ q_rcx ~Å~  q_a; 

 

So now we have five of category-symbol q_cx or q_xc, and, again, seven of category-symbol q_r or q_cc or q_xcx, and so on.”

 

“How do we escape 5q_cx and 7q_r? 

Answer: Again, Axiom §7.”

 

“How do we avert losing interpreted/applied ontic category-symbols q_x, q_c, q_cx, and q_r, which corresponding, in the generic/unapplied/-minimally-interpreted _NQ dialectical arithmetic to its first four ‘meta-numerals’ – 

q₁, q₂, q₃, and q₄, respectively? 

 

Answer: Again, Axiom §9.”

 

“Finally, how do we get from the four rows and thirty-two category-symbols of the “raw” product for the third-epoch, with q_x raised to the power 8?”

“The key is to be consistent about our solution [‘|-=’] for each repeat subscript category-symbol, and for the ordinal number corresponding to the subscript(s) of each distinct category-symbol, resolving upon the same set of subscripts for all occurrences of category-symbols undergirded by that ordinal number in the undergirding, minimally-interpreted, generic _NQ arithmetic, which has only ordinal-numbers as its subscripts.” 

 

“For example, the ordinal number values corresponding to the subscripts of category-symbols q_x [|x| = ordinal 1];

q_xx |-= q_c [|c| = |xx| =  |x|+|x| =

1+1 = ordinal 2];

q_cx [|cx| = |c|+|x| = 2+1 = ordinal 3], 

and;

q_cc |-= q_r [|cc| = |c|+|c|  = 2+2 = ordinal 4]

– are –

1^st;

2^nd = 1^st plus 1^st;

3^rd = 2^nd plus 1^st, and;

4^th = 2^nd plus 2^nd,

respectively.”

 

“Per these considerations, the progression of subscript commutations and consistent double-subscript designations/solutions that gets us to the eight, mutually opposing [‘~’] – as well as gapless, via Axiom §9, and ‘de-redundantized’, via Axiom §7 – category symbols –

q_x ~Å~ q_c ~Å~ q_cx ~Å~ q_r ~Å~      

q_rx ~Å~ q_rc ~Å~ q_rcx ~Å~  q_a;

– are the following –

(q_xÅq_xx)Å(q_cÅq_xc)Å(q_cxÅq_xcx)Å(q_rÅq_xr) 

Å  

(q_xÅq_cx)Å(q_cÅq_cc)Å(q_cxÅq_ccx)Å(q_rÅq_cr) 

Å  

(q_xÅq_cxx)Å(q_cÅq_cxc)Å(q_cxÅ q_cxcx) Å

(q_rÅ q_cxr) 

Å  

(q_xÅq_rx)Å(q_cÅq_rc)Å(q_cxÅq_rcx)Å(q_rÅq_rr) 

– transforms to –

(q_xÅq_xx)Å(q_cÅq_cx)Å(q_cxÅq_cxx)Å(q_rÅq_rx) 

Å  

(q_xÅq_cx)Å(q_cÅq_cc)Å(q_cxÅq_ccx)Å(q_rÅq_cr) 

Å  

(q_xÅq_cxx)Å(q_cÅq_xcc)Å(q_cxÅq_ccxx) Å

(q_rÅq_rcx) 

Å  

(q_xÅq_rx)Å(q_cÅq_rc)Å(q_cxÅq_rcx)Å(q_rÅq_a)

– which next transforms to –

(q_xÅq_c)Å(q_cÅq_cx)Å(q_cxÅq_cc)Å(q_rÅq_rx) 

Å  

(q_xÅq_cx)Å(q_cÅq_cc)Å(q_cxÅq_ccx)Å(q_rÅq_cr) 

Å  

(q_xÅq_cc)Å(q_cÅq_xr)Å(q_cxÅq_ccc) Å

(q_rÅq_rcx) 

Å  

(q_xÅq_rx)Å(q_cÅq_rc)Å(q_cxÅq_rcx)Å(q_rÅq_a)

– which next transforms to, with now-reveled redundant category-symbols terms crossed-out –

(q_xÅq_c)Å(q_cÅq_cx)Å(q_cxÅq_r)Å(q_rÅq_rx) 

Å  

(q_xÅq_cx)Å(q_cÅq_r)Å(q_cxÅq_rx)Å(q_rÅq_rc) 

Å  

(q_xÅq_r)Å(q_cÅq_rx)Å(q_cxÅq_rc) Å

(q_rÅq_rcx) 

Å  

(q_xÅq_rx)Å(q_cÅq_rc)Å(q_cxÅq_rcx)Å(q_rÅq_a)

– which arrives at –

q_x~Å~q_c~Å~q_cx~Å~q_r~Å~q_rx~Å~  

q_rc~Å~  

q_rcx ~Å~  

q_a.”

 

 

 

 

 

 

 

 

 

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.

 

 

 

 

 

 

YOU are invited to post your comments on this blog-entry below!