Saturday, January 04, 2014

Part 2 of Recent Internet Dialogue on Seldonian Dialectical Algebra.

Dear Readers,



Below I have posted an edited excerpt from the second part of a recent web forum thread dialogue on F.E.D. dialectics in which I participated.

 
I have once again notated my interlocutor's queries by 'Q#.:', and my responses by 'R#.:'.

 
Please feel free to join in this dialogue here, by posting a query or comment via the Comment function of this blog.



Regards,

Miguel




 




 

Q2.:


thesis x antithesis = synthesis.


I get that.


But then what?


What can be derived from this??






R2.: From this, we can construct a Universal Method, called ‘‘‘The Dialectical Method’’’, whereby theories -- explaining the pasts, and predicting the futures -- of the “totalities”, i.e., of the “systems” of phenomena, that we encounter and experience, can be, first, discovered, and then presented systematically, and then mnemonically retained / remembered, via an ultimate degree of knowledge representation compression, that is readily -- intuitively and algorithmically -- re-expandable.

Discovering and then sharing/disseminating such a dialectical-scientific theory, of/for such a totality, involves discovering and organizing the categories that can “comprehend” the experienced, empirical content of that totality, i.e., that can comprehensively organize the entire content of all of the experiential, phenomenal entities that make up that totality, in terms of the various qualities of the various sub-groups of those entities, and discovering the progression or progressive order of those categories -- how those categories tie together; how one category “begets” another, in a kind of ‘categorial meta-genealogy’-- and then presenting those categories, presenting their categorial progression, i.e., their meta-genealogy -- orally, and/or in writing, systematically.

By such a systematicpresentation, I mean one which, presents the categories that “comprehend” the given totality, starting from the simplest category of the given totality, then moving on to present the next more complex category, then to the next, still more complex category after that -- in monotonic order of rising complexity -- until the richest, most explanatorily powerful, most complex category “comprehending” that totality is reached/achieved, and presented, which should also be the intuitive order of the connexions between one category and the next.

The Seldon arithmetic and algebra of for dialectics gives heuristic, algorithmic support to these intuitive processes of theory discovery/formation, theory presentation, and mnemonic theory retention. Here’s how.


To discover/present/retain such dialectical-scientific theories, for most of the key totalities, we need, first, to get beyond a mere three categories, i.e., beyond --

thesis category + antithesis category + synthesis category

-- to as many categories, in dialectical categorial progression, as we need to “comprehend” / explain the given totality.

So let us broaden our terminology, to accommodate names for more than just the first three categories of generic dialectics.

For the “dyadic” Seldon function version of the general dialectic, the fundamental rules are the following --

thesis x thesis = thesis + antithesis

antithesis x thesis = thesis + synthesis
-- which we can broaden, to accommodate any number of additional, higher antitheses and syntheses, as follows:

first thesis x first thesis = first thesis + first antithesis

first antithesis x first thesis = first thesis + first synthesis.


So that we get a progression, by “squaring” each result, i.e., by “interacting” every component of our result with itself, and with every other components), that looks like this --

point-of-departure: first thesis

first thesis squared with itself = first thesis x first thesis = first thesis + first antithesis;

(first thesis + first antithesis ) squared = (first thesis + first antithesis) x (first thesis + first antithesis) =

(first thesis x first thesis) +
(first thesis x first antithesis) +
(first antithesis x first thesis) +
(first antithesis x first antithesis) = [highlighting the new]:

(first thesis + first antithesis) +
(first antithesis + first synthesis) +
(first thesis + first synthesis) +
(first antithesis + second antithesis)

-- which, taking out parentheses, becomes --

first thesis + first antithesis + first antithesis + first synthesis + first thesis + first synthesis + first antithesis + second antithesis.


Now, removing redundant occurrences of generic dialectical categories from the above, i.e., removing all but the first appearance of each category, we get --

first thesis + first antithesis + first synthesis + second antithesis.


If we again “square” this result with itself, and again eliminate redundant occurrences of categories, we get --

(first thesis + first antithesis + first synthesis + second antithesis) squared =

(first thesis + first antithesis + first synthesis + second antithesis) x (first thesis + first antithesis + first synthesis + second antithesis) =

first thesis + first antithesis + first synthesis + second antithesis + first partial synthesis + second partial synthesis + second synthesis + third antithesis.


We can broaden are terminology again, to make the whole terminology jibe with the emergence of “partial syntheses” as follows --

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis.





If we think of the “first_thesis”, or “thesis_#1”, as representing, in effect “full_synthesis_#0”, then the following system of two, coupled, dialectical-algebraic equations is general for the dyadic Seldon function version of generic or general dialectics -- i.e., for dyadic Seldon function generated, generic dialectical categorial progressions --


full_antithesis_#N x full_antithesis_#N = full_antithesis_#N + full_antithesis_#N+1

&

full_antithesis_#N+1 x full_synthesis_#N = full_synthesis_#N + full_synthesis_#N+1.





[The corresponding system of two, coupled, dialectical-algebraic equations for the general triadic Seldon function version of generic or general dialectics -- i.e., for triadic Seldon function generated, generic dialectical categorial progressions -- is the following:


full synthesis_#N x full synthesis_#N = (full synthesis_#N)^2 =

full synthesis_#N + full_antithesis_#N+1

&

full_antithesis_#N+1 x full_synthesis_#N = full_synthesis_#N + full_synthesis_#N+1.




The expansion of the generic-dialectical triadic Seldon function through its first nine terms is --


first_thesis^(3^2)  = first_thesis^9 =



first_thesis + first full antithesis + first full synthesis + first partial antithesis + second partial antithesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis ].







I should note, here, that we often find ourselves moved to use what seem to us to be more pertinent generic dialectical category names than those involving “thesis”, “antithesis”, and “synthesis”, when expressing the content stated otherwise above.

The terms thesis, “antithesis, and “synthesis bespeak “theses” as propositions -- e.g., as in the exchange of propositions and of counter-propositions [and of synthesizing ‘uni-propositions’] in a dialogue among two or more voices/persons, or as in the internal, e.g., inaudible, unspoken self-dialogue’ of the thought-process within the mind of a single person.




Indeed, such were the historical origins, and contexts, and of the original meanings, and of the etymology, of the word «dialektikê».






However, for the context of systematic dialectics” -- i.e., of systematically-ordered categorial progressions, describing and explaining the present organization and reproduction of a given totality, we find the following nomenclature more pertinent:

chosen starting category + first full contra-category + first full uni-category + second full contra-category + first partial uni-category + second partial uni-category + second full uni-category + third full contra-category . . ..


On the other hand, for the context of historical dialectics” -- i.e., of chronologically-ordered categorial progressions, describing /- reconstructing, and explaining, the past historical progression of a whole historical sequence of different, successive physical totalities, especially for pre-human, ‘pre-mental’, ‘‘‘pre-cognitive’’’, ‘pre-«psychê»-ic’ totalities of the overall dialectic of Nature, we find the following nomenclature more pertinent, given the meaning of the ancient Greek word -- the fact, the principle, the pattern of self-growth in Nature:

chosen starting «physis» + first full meta-«physis» + first full uni-«physis» + second full meta-«physis» + first partial uni-«physis» + second partial uni-«physis» + second full uni-«physis» + third full meta-«physis». . ..




Whereas, for the context of historical dialectics” involving human, ‘‘‘mental’’’, ‘‘cognitive’’, ‘«psychê»-ic’ totalities of the overall dialectic of Nature, and building on the word ‘‘‘meme’’’, we find the following nomenclature more pertinent, given the meaning of the ancient term ‘‘‘meme’’’, and forming, from it, the neologism memesis to name the expression of any system of memes, all the way up to and including a cultural totality, an historically specific cultural/ideological “superstructure”, an epoch in the ‘meta-evolution’ of ‘Phenome’ of the total human species:

chosen starting memesis + first full contra-memesis + first full uni-memesis + second full contra-memesis + first partial uni-memesis + second partial uni-memesis + second full uni-memesis + third full contra-memesis. . ..





We can always go on, if need be, by again squaring these sums of eight categories, variously ‘nomenclatured’ above, each with itself, and by then re-squaring that squaring, and so on, to some indefinite ultimate extent, e.g., until we have all of the generic dialectical categories we need for the given specific dialectical modeling application thereof.



But, instead, let’s stay with this eight-category stage of generic dialectics, and consider some specific examples of this generic dialectic.





Example 1. Let’s consider, first, the classic example of scientific dialectics, the theory, and the presentation of the theory, of capitalism, in Marx’s magnum opus, Capital: A Critique of Political Economy. In that work Marx is not, primarily, presenting the historical dialectic leading from primitive communism, through the progression of class societies, ending with capitalism, and on to higher, democratic communism [as distinguished from, e.g., Stalinist, plutocratic state-capitalism]. Instead, he is taking a cross-section of historical time, the present representative “slice” of history, that contains only contemporary capitalist class society, e.g., mainly in the U.K. of his time, and presenting, systematically, the categories that account for/explain the present, temporary self-reproduction of this capitalist system.

Let us limit ourselves to the forms of [exchange-]value categories that Marx includes, and that manifest primarily in the circulation process of capitals, not trying to cover, in this initial “dialectical model” of the content of Marx’s treatise, the categories that manifest primarily in the production process of capitals, in order to keep this first example model a little simpler than it would otherwise have to be..

These categories of capitalist class society only encounter or encompass the past slice(s) of historical time, that contained its predecessor system(s) of class society, before that capitalist slice of history, and the predicted future slice(s) of historical time, after the predicted, self-non-reproduction, or self-«aufheben» self-negation, of the capitalist system, only at the two, opposite margins/ends of their representative “slice” -- only at the beginnings and at the ends of their categorial progression -- of the categorial progression that comprehends capitalism / the capitalist “slice” of history.


This kind of dialectics is called “systematic dialectics”, or ‘synchronic dialectics’, as [dialectically] opposed to “historical dialectics”, or ‘diachronic dialectics’.


Marx divides his work into an overall thesis category + antithesis category + synthesis category structure --


thesis: vol. I -- “The Productions-process of Capitals” [German: «Der Produktionsprocess des Kapitals»], qP;

antithesis: vol. II -- “The Circulations-process of Capitals” [German: «Der Cirkulationsprocess des Kapitals»], qC;

synthesis: vol. III -- ‘‘‘The Process of Capital-Reproduction as a Whole’’’, qCP


-- a dialectic whereby we can symbolize the content of the whole work as --


qP + qC + qCP


-- wherein ‘q’ stands for a qualitative category, by way of the quality, shared by all of the entities or individual things/phenomena “comprehended” by any given category, and wherein qP stands for the totality of capitalist “Production” phenomena, and wherein qC stands for the totality of capitalist “Circulation” phenomena, and wherein qCP stands for the phenomena which combine and bind and integrate the Circulation and the Production phenomena together, into a “complex unity” or “synthesis” -- into a whole -- which constitutes the “Capitalist System” as a totality, and wherein the underscore, _, under the q, signifies the ‘contra-Boolean’ character of the arithmetical, algorithmic behavior of qX, whatever particular quality-name the symbol ‘X’ may abbreviate, i.e., the ‘contra-Boolean’ character of the addition and multiplication behaviors of qX.


Suppose we therefore take the category of “Commodities” -- from Capital, volume I., Part I. [entitled “COMMODITIES AND MONEY”], Chapter I., entitled, simply, “Commodities”, as our special case of the generic “first thesis”.

We then have the following application of generic dialectics to model our specific part of dialectic of Capital:

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis

-- interpreted / specified in terms of the ‘“value-forms”’ Marx’s Capital theory --

commodities + monies + monies-mediated Circulations of commodities + «kapitals» + commodity-«kapitals» + money-«kapitals» + The Circulations of the Total Social «kapital» + portents of the developing Dialectical Self-Negation of the «kapitals»-System

-- or, in our “shorthand” symbols --

qc + qcc [ = qm ] + qmc + qmm [ = qk ]+ qkc + qkm + qkmc + qkk =

qc + qm + qmc + qk + qkc + qkm + qkmc + qkk

-- such that --

(1) the first thesis category symbol, qc, stands for the multitude of individual commodities that inhere in this, our present, Capitalist, social totality, and for the content of Capital, volume I, Part I, Chapter I.;

(2) the first full antithesis category symbol, qm, stands for the multitude of individual units of monies that inhere in our present social totality, and for the content of Capital, volume I, Part I, Chapter I., Section 3., D., wherein monies are grasped, ‘meta-genealogically’, as qcc -- as ‘meta-commodities, each one made up out of a heterogeneous multiplicity of commodities -- as a mental list, or meme, of the kinds of commodities whose owners will exchange them for monies [not a universal list at the earlier stages of commodity/money confrontation, and not a universal list of all “commodities” even today!];

(3) the first full synthesis category symbol, qmc, stands for the multitude of individual transactions, involving both Monies and Commodities, combined, as a “complex unity” process of “the monies-mediated circulations of commodities”, that inhere in our present social totality, and that Marx symbolized as C-M-C’, and for the content of Capital, volume I, Part I, Chapter III., Section 3., “...the Circulation of Commodities”;

(4) the second full antithesis category symbol, qk, stands for the multitude of individual «kapitals» that inhere in our present, Capitalist, social totality, and for the content of Capital, volume I, Part II, Chapter IV., “...the Circulation of Commodities”, whereby «kapitals» are grasped, ‘meta-genealogically’, as qmm -- as ‘meta-monies, each one made up out of a multiplicity of monies -- as a mental listing / accounting record / accounting sum meme, of the net profits, or retained earnings, and of the other net current assets [net “Equity”] over the whole history, to-date, of the past accounting periods, or aggregate transactions, during the existence of each given individual «kapital» thereby represented;

(5) the first partial synthesis category symbol, qkc, stands for the multitude of those portions of individual units of «kapital» that presently, temporarily take the form of multitudes of commodities, thus units of commodities and of «kapitals» combined, as a “complex unity”, named, by Marx, “Commodity-Capital”, that inhere in our present society, and that constitute a partial synthesis category because qkc unifies the categories of qk and qc, but without also unifying them with the other already extant/already presented category, qm. The symbol qkc also stands for the content of Capital, volume II, Part I, Chapter III., “the Circuit of Commodity-Capital”;

(6) the second partial synthesis category symbol, qkm, stands for the multitude of portions of individual units of «kapital» that presently, momentarily take the form of multitudes of monies, thus units of monies and of «kapitals» combined, as a “complex unity”, named, by Marx, “Money-Capital”, that inhere in our present society, and that constitute a partial synthesis category because qkm unifies the categories of qk and qm, but without also unifying them with the other already extant/already presented category, qc. The symbol qkm also stands for the content of Capital, volume II, Part I, Chapter I., “the Circuit of Money-Capital”;

(7) the third full synthesis category symbol, qkmc, stands for the multitude of presently existing portions of each individual unit of «kapital» that presently, transiently take the form of a multitude of money units, qkm, on the way to ‘metamorphosizing’ into a multitude of units of commodities, qkc, on the way toward then re-‘metamorphosizing’ into a multitude of units of Monies, qkm, and for the multitude of presently existing portions of each individual unit of «kapital» that presently, transiently take the form of a multitude of individual commodity units, qkc, on the way to ‘metamorphosizing’ into a multitude of money units, qkm, on the way toward then re-‘metamorphosizing’ into a multitude of units of commodities, qkc, but combined, as a complex unity, named, by Marx, ‘“The Circulations-Process of the Total Social Capital”, by which kapital, qk, appropriates, to itself, the “money mediated circulation of commodities” , qmc, and, equally, the “commodities-mediated circulation of monies” , qcm -- via, e.g., qkmc = qk x qmc, minus qmc -- that inheres in our present society, and that constitutes a full synthesis category because qkmc unifies all three categories -- qk, qm, and qc -- that are already extant/already presented, so far, in this categorial progression. The category-symbol qkmc also stands for the content of Capital, volume II, Part III, “THE REPRODUCTION AND CIRCULATION OF THE AGGREGATE SOCIAL CAPITAL;

(8) the third full synthesis category symbol, qkk, stands for the multitude of processes of interaction among different individual «kapitals» -- that drive the competition of capitals, the expropriation of many «kapitals» by single «kapitals», the consolidation, concentration, and centralization of «kapitals», the accumulation of «kapitals», the rise in the organic composition of «kapitals», the tendency of the rate of profit on those «kapitals» to fall, and global economic crises of increasing severity -- that inhere in our present, Capitalist, social totality, and that drive its internal evolution toward the limits of the «kapitals»-system, and towards the possible birth, from out of its womb, of a higher human social system/totality, and thus also portending the emergence of some kind of ‘meta-«kapital(s)»’ [ev]entity. The category-symbol qkk also stands for the content of Capital, volume I, Part VII, “THE ACCUMULATION OF CAPITAL”, Chapter XXV., Section 3., “Progressive Production of a Relative Surplus-Population or Industrial Reserve Army”, of Part VII, “THE SO-CALLED PRIMITIVE ACCUMULATION”, Chapter XXXII., “Historical Tendency of Capitalist Accumulation”, of volume III, Part III, “THE LAW OF THE TENDENCY OF THE RATE OF PROFIT TO FALL”, and of volume III, Part V, Chapter XXVII., “The Role of Credit in Capitalist Production”.


[For those who understand “exponents” or “powers” of algebraic variables:]

This whole categorial-progression representation of Marxian knowledge can be condensed massively into just ten symbolic elements --

(qc)^(2^3), or into just six symbolic elements (qc)^8, or into just four symbolic elements c^8

-- the latter meaning ‘commodities^8’, in a way that can be re-expanded intuitively -- ‘re-minded’ -- into the full progression using the algorithm we have described above.





Reality is "everywhere dense" with dialectic.

Dialectical, [dynamical 'qualo-fractal'] 'content structures' are ubiquitous in Nature as a whole, and, '''holonomically''', independently, in its parts as well, in both re-constructed, past pre-human Nature, and in contemporary, extra-human Nature, as well as within the "artefacts" produced, and emerged within, and still emergent within, contemporary '''human Nature''', within 'The Human Phenome' of today.




Example 2. For the system of, e.g., English phonetic written language, starting category = letters [of the alphabet], we have the following application of generic dialectics to model the specific dialectic for this domain:

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis

-- interpreted / specified in terms this specific domain --

letters + words + exceptions: single-letters-that-are-also-words [e.g. 'a' and 'I' [in Spanish, e.g., 'y']] + sentences + exceptions: single-letter-sentences + exceptions: single-word-sentences + exceptions: single-letters-that-are-also-single-words-that-are-also-sentences + paragraphs

-- or, in our “shorthand” symbols --

qL + qLL [ = qW ] + qWL + qWW [ = qS ]+ qSL + qSW + qSWL + qSS [ = qP ] =

qL + qW + qWL + qS + qSL + qSW + qSWL + qP
;



Example 3. For the system of, e.g., the English time metrology, starting category = seconds:

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis

-- interpreted / specified in terms this specific domain --

seconds + minutes + intermediate durations, expressed in terms of both minutes and seconds + hours + intermediate durations, expressed in terms of hours plus seconds + intermediate durations, expressed in terms of hours plus minutes + intermediate durations, expressed in terms of hours plus minutes plus seconds + days

-- or, in our “shorthand” symbols --

qS + qSS [ = qM ] + qMS + qMM [ = qH ]+ qHS + qHM + qHMS + qHH [ = qD ] =

qS + qM + qMS + qH + qHS + qHM + qHMS + qD
;



Example 4. For the domain of contemporary TV Series, starting category = Episodes:

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis

-- interpreted / specified in terms this specific domain --

Episodes + Seasons + exceptional, single Episode [e.g., cancelled early] Seasons + aeRies + exceptional, single Episode seRies + single Season seRies + exceptional, single Season seRies which are also single Episode seRies + ‘Multi-seRies’

-- or, in our “shorthand” symbols --

qE + qEE [ = qS ] + qSE + qEE [ = qR ]+ qRE + qRS + qRSE + qRR [ = qM ] =

qE + qS + qSE + qR + qRE + qRS + qRSE + qM
;



Example 5. For the system of, e.g., American coinage, starting category = pennies:

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis

-- interpreted / specified in terms this specific domain --

pennies + nickels + intermediate monetary amounts, expressed in terms of both nickels and pennies + dimes + intermediate “change” amounts, expressed in terms of dimes plus pennies + intermediate amounts, expressed in terms of dimes plus nickels + intermediate amounts, expressed in terms of dimes plus nickels plus pennies + quarter[-dollar]s

-- or, in our “shorthand” symbols --

qP + qPP [ = qN ] + qNP + qNN [ = qD ]+ qDP + qDN + qDNP + qDD [ = qQ ] =

qP + qN + qNP + qD + qDP + qDN + qDNP + qQ
;



Example 6. For the system of “positive”, or ‘proverse’, operations of ordinary arithmetic, starting category = addition:

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis

-- interpreted / specified in terms this specific domain --

additions + multiplications + “distributions” of multiplications over additions [and of additions over multiplications] + exponentiations + exponentiations over additions + exponentiations over multiplications + exponentiations over multiplications and additions + ‘meta-exponentiations’ [e.g., the 3 in (qc)^(2^3) is a ‘meta-exponent’]

-- or, in our “shorthand” symbols --

qA + qAA [ = qM ] + qMA + qMM [ = qE ]+ qEA + qEM + qEMA + qEE =

qA + qM + qD + qE + qEA + qEM + qEMA + qEE
;



Example 7. For the system of “negative”, or ‘inverse’, operations of ordinary arithmetic, starting category = subtraction:

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis

-- interpreted / specified in terms this specific domain --

subtractions [‘de-additions’] + divisions [‘multi-subtractions’; ‘de-multiplications’] + “combinations of division(s) and subtraction(s) + root-extractions [‘divisions of divisions’; ‘de-exponentiations’] + combinations of root extractions and subtractions + combinations of root-extraction(s) and division(s) + combinations of root-extraction(s) with both division(s) and subtraction(s) + ‘de-meta-exponentiations’ [e.g., the operation that ‘‘‘extracts’’’ 3 from (qc)^(2^3) is a ‘de-meta-exponentiation’]

-- or, in our “shorthand” symbols --

qS + qSS [ = qD ] + qDS + qDD [ = qR ]+ qRS + qRD + qRDS + qRR =

qS + qD + qDS + qR + qRS + qRD + qRDS + qRR
;




Example 8. For the system of, e.g., modern ‘computerware’ -- with the word ‘computerware’ standing for the category of the synthesis, or “complex unity”, of the categories “computer hardware” and “computer software” -- starting category = [binary] bits, we have the following application of generic dialectics to model the specific dialectic for this domain:

first thesis + first full antithesis + first full synthesis + second full antithesis + first partial synthesis + second partial synthesis + second full synthesis + third full antithesis

-- interpreted / specified in terms this specific domain --

bIts + bYtes + exceptions: one-bit bytes + computer words + exceptions: single-bit computer words + exceptions: single-byte computer words + exceptions: single-byte-computer-words-that-are-also-single-bit-computer-words + computer-language commands

-- or, in our “shorthand” symbols --

qI + qII [ = qY ] + qYI + qYY [ = qW ]+ qWI + qWY + qWYI + qWW [ = qC ] =

qI + qY + qYI + qW + qWI + qWY + qWYI + qC
.





Next, I’ll give you some examples of dialectical-algebraic models for historical dialectics. . . ..



How much of this is clear to you, so far?

Please lodge additional queries as needed!

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