Saturday, September 28, 2013

Seldonian Dialectical Algebra Simplified.


Seldonian Dialectical Algebra Simplified.





Dear Readers,

One of my internet interlocutors recently wrote:  “. . . I can't make heads or tails out of the algebra”.

¡Here’s hoping that the simplified summary below will reveal for you here both the “heads” and the “tails” of Seldon’s sensational discovery!


Regards,

Miguel







Preliminaries.  Suppose ‘~’ stands for the operation of dialectical negation generically, and that this generic ‘~’ comes in many specific, determinate varieties, ‘~x’, where the subscript category-variable, x, is the "determination" of the dialectical negation operation, i.e., where x specifies the variety of the negation ‘categorially’ -- as opposed to ‘¬’, standing for formal-logical negation, generically only.  

If so, by the rules of the Seldonian dialectical algebra of the NQ dialectical arithmetic, specifically, with a and b representing qualitatively distinct “ontological” categories -- kind of thing” categories -- then:  

a x b   =  ~a[ b ], and, in particular: 

a x a   =  ~a[ a ].

That is, the “times” operation, or ontological-categorial “multiplication” operation, is interpreted, in Seldonian dialectical algebra, as representing dialectical, determinate negation, of the multiplicand -- b, in our first example -- by its multiplier, a, and, also, thereby, the “time” operation as well -- as ‘time generation’, the generation of historical order-of-progression, or of the time/order-of-presentation of a step in the systematic presentation of the content of a “synchronic slice”, or “snapshot”, representing an historical present. 

In the presentation(s) / narratives below, the names of dialectical categories -- of generic categories of the Marxian dialectic -- are indicated by underscoring those names.

The arrow symbol, ‘--->’, indicates the process of historical becoming, or of a systematic shift in presentational focus, and the direction of the arrow indicates that direction of becoming / shifting.

Exponentiation is signified via the '^' sign, e.g., 3^2  =  3 "squared"  =  9.

Two different, generic dialectical interpretations of the Seldonian arithmetic/algebra are presented below, the first, dyad-ic’, the second, triad-ic’.



A.  Seldonian Dialectical Algebra, Dyad-icInterpretation.



Given:  The initial, ‘Stage 0’ category, designated as the generic dialectical “thesis” category, including the shared, implicit “connotations” of that category, assigned [‘[---]’] to the generic Seldonian ‘1st dialectical meta-number’, q1 --  

q1  [---]  thesis.



Stage 0:  thesis  =  thesis^1.



Stage 1:  “Square” the thesis, the result of Stage 0, with itself --


thesis^2  =  thesis^1  x  thesis^1  =   

thesis^1[ thesis^1 ]     =   

thesis[ thesis ]   =   

~thesis[ thesis ]  =  


thesis "of" thesis  =


thesis + antithesis;  


this qualitatively heterogeneous categories-sum of two ontological categories, here described only generically, as “thesis” and/versus as “antithesis”, does not “simplify” to any single category, precisely because these two categories are mutually inhomogeneous; qualitatively different, like the proverbial “apples+oranges”.  

Overall, we can describe this operation, of ‘ontological multiplication’, connoting also one of 'ontological [self-]proliferation' -- of the ‘dialectical [self-]multiplication’ -- of the category [ thesis ] as operand/argument/multiplicand, by itself, by that category thesis[ ... ], again, this time as operator/function/multiplier, as a «aufheben» double-conservation of the operand, [ thesis ], including by adding that operand back in the result/function-value/product, as well as by the conservation/containment-negation of the thesis in the antithesis: 

t x t   =  t[ t ]   =  ~t[ t ]  =  t + deltat ]   =   t  +  a.



Stage 2:  “Square” the result of Stage 1 with itself --


[thesis + antithesis]^2   =    [thesis + antithesis]  x  [thesis + antithesis]   =


[thesis + antithesis]  +  

[ [thesis x thesis ] + [thesis x antithesis] + [antithesis x thesis] + [antithesis x antithesis] ]   =


[thesis + antithesis]  +  

[ [thesis + antithesis ] + [thesis + synthesis] + [antithesis + synthesis] + 

[antithesis + second antithesis] ]  =


thesis + antithesis + thesis + antithesis + thesis + synthesis + antithesis + synthesis + antithesis +

second antithesis; 


The thesis category occurs 3 times in the foregoing heterogeneous sum of categories, the antithesis category 4 times, and the synthesis category occurs twice.  

But just a single occurrence of each unique, qualitatively distinct ontological category is sufficient; more than that is redundant.  

So that sum, after we apply to it of the rule of single occurrence sufficiency just articulated, simplifies to just four terms:


thesis + antithesis + synthesis + second antithesis.


‘Retro-renaming’ the first 3 terms to cohere with the naming of the 4th term yields:  

first thesis + first antithesis + first synthesis + second antithesis


Using the symbol ‘---> to stand for the phrase “goes to”, or for the word “becomes”, we obtain the following re-rendition of the whole story of this section --



Re-Rendered in Phono-Ideo-gram-icalShorthand: 

t1  =  t1^1,  --->  t1^2 = t1 + a1 = [t1 + a1]^1,  --->  [t1 + a1]^2 = t1 + a1 + s1 + a2.



Summary:  Dyadic Seldon Function,

2|-|-|h  =  [ t1 ]^(2^h)


for h = 0, 1, 2.



Next Stage:  Stage 3, h = 3 -- 

2|-|-|3   =   [ t1 ]^(2^3)   =  

[ t1 ]^(8)   =   t1 + a1 + s1 + a2 + ps1 + ps2 + s2 + a3

in which “ps” stands for “partial synthesis”. 

¡If the examples above have “given you the gist” of how the Seldonian rules work, then “do the math yourself” for Stage(s) 3(+), and see what you get!






BSeldonian Dialectical AlgebraTriad-ic’ Interpretation.


Given: The initial, ‘Stage 0’ category, designated as the generic dialectical “thesis” category, including the shared, implicit “connotations” of that category, assigned [ ‘[---]’ ] to the generic Seldonian ‘first dialectical meta-number’, q1 --

q1 [---] thesis.


A “cool” thing about Seldon’s algebra -- 

q1^n   =   q1  + ... +  qn

-- with '+ ... +', as opposed to mere '+ ... +', meaning the sum of all of the q 
numbers, with every consecutive “natrual” number subscript between 1 and n [inclusive].


Stage 0:  thesis  =  thesis^1.

Stage 1:  dialectical self-negation of the dialectical self-negation of the thesis

the result of Stage 0:


thesis^3 =

thesis^1 × thesis^1 × thesis^1 =

thesis^1[ thesis^1[ thesis^1 ] ] =

thesis[ thesis[ thesis ] ] =

thesis
"of" [ thesis "of" thesis ] = 

~thesis~thesis[ thesis ] ] = 

thesis × thesis × thesis,

which involves two distinct ‘associativities’, or orders of multiplicative operation,

thesis × [ thesis × thesis ],

versus

[ thesis × thesis ] × thesis,

addressed below in (1.a.) and (1.b.), respectively,

to show that both converge to the same result.


(1.a.) thesis^3  =  thesis × [ thesis × thesis ], =

thesis × [ thesis + antithesis ], =

[ thesis + antithesis ] + [ thesis × thesis ] + [ thesis × antithesis ], =

[ thesis + antithesis ] + [ thesis + antithesis ] + [ antithesis + synthesis ], =

thesis + antithesis + thesis + antithesis + antithesissynthesis,

which, after removing “redundant” repetitions of categories / terms, simplifies to:

thesis + antithesis + synthesis.


Re-Rendered in ‘Phono-Ideo-gram-ical’ Shorthand:

t1 = t1^1, t1^3 =

t1 × [ t1 × t1 ] =

t1 × [ t1 + a1 ] =

[ t1 + a1 ] + [ t1 × t1 ] + [ t1 × a1 ] =

[ t1 + a1 ] + [ t1 + a1 ] + [ a1 + s1 ] =

t1 + a1 + t1 + a1 + a1s1 =

t1 + a1 + s1
.


(1.b.) thesis^3 = [ thesis × thesis ] × thesis, = 

[ thesis + antithesis ] × thesis, =

[ thesis ] + [ thesis × thesis ] + [ antithesis × thesis ], =

[ thesis ] + [ thesis + antithesis ] + [ thesis + synthesis ], =

thesis + thesis + antithesis + thesis + synthesis,

which, after removing “redundant” repetitions of categories / terms, simplifies to:

thesis + antithesis + synthesis.


Re-Rendered in ‘Phono-Ideo-gram-ical’ Shorthand:

t1 = t1^1, t1^3 =

[ t1 × t1 ] × t1 =

[ t1 + a1 ] × t1 =

t1 + [ t1 × t1 ] + [ a1 × t1 ] =

t1 + [ t1 + a1 ] + [ t1 + s1 ] =

t1 + t1a1 + t1s1 =

t1 + a1 + s1
.


SummaryTriadic Seldon Function, 3|-|-|h = [ t1 ]^(3^h), for h = 01.


Next StageStage 2, h = 2 -- 

3|-|-|2   =   [ t1 ]^(3^2)   = 

[ t1 ]^(9)   = 

t1 + a1 + s1 + pa1 + pa2 + a2 + ps1 + ps2 + s2

in which “pa” stands for “partial antithesis”.

¡If the examples above have “given you the gist” of how the Seldonian rules work, then “do the math yourself” for Stage(s) 2(+), and see what you get!



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