I want to provide you with a simplified account of the F.E.D. [Foundation Encyclopedia Dialectica] "First Dialectical Arithmetic", and of its "Algebra" [which differs in special ways from standard algebras which are purely-quantitative, or which are otherwise not explicitly dialectical].
If you want the full-regalia, complex account of these matters, you'll have to go to the F.E.D. Primers.
Suppose we take q to denote a generic symbol, an "order qualifier", for qualitative "ordinality", or "ordered-ness" in general; for the "genus" of "consecutive, discrete ordered-ness".
Suppose we take q/1 to denote the specific ordinal quality "species" of "first-ness", and to denote also the "purely-qualitative division" of "purely-quantitative" Natural Number 1 into the "purely qualitative" generic "order qualifier" q [called "qualitative" division because numerator and denominator differ in quality, so that the division does not result in any "reduction", or "amalgamation" to a less-complex value ].
Since q and 1 are mutually qualitatively different, or heterogeneous, q/1 is a division, or fraction, which does not reduce.
Suppose also that we take q/2 to denote another non-reductive division/fraction, and to denote the specific ordinal quality species of "second-ness".
Suppose as well that we take q/3 to denote yet a third "ordinal qualifier", denoting yet another -- the next -- specific ordinal quality species, the quality of "third-ness".
Then the generic q/1, q/2, and q/3 can be applied as symbols for generic ontological qualifiers, standing for qualitatively distinct ontological categories, assignable to more specific successive terms in a "dialectical [meta-]model", according to the following "dialectical interpretation" of these three purely-qualitative "meta-numerals", formed by this aufheben negation / conservation / elevation of the purely quantitative numerals 1, 2, and 3, respectively:
q/1 -- assign to the "first, or arche', thesis" of the first triad of a specific dialectical progression / succession / "consecuum";
q/2 -- assign to the "first anti-thesis", or "first contra-thesis", of the first triad of a specific dialectical progression / succession / "consecuum";
q/3 -- assign to the "first [full] synthesis", or "first [full] uni-thesis", of the first triad of a specific dialectical progression / succession / "consecuum". The epithet "full" means, in this context, that synthesis encompasses all previously-posited ontological categories posited as possibly having instantiations existent, or "extant".
Thereafter, q/4, q/5, q/6, q/7, etc., can be applied to denote, e.g., the second "contra-thesis", the "first partial uni-thesis", the "second partial uni-thesis", and the "second full uni-thesis", respectively, etc.
Given the rules [axioms] for "ontological qualifier" "addition" and "multiplication", the formula that F.E.D. calls "The Dyadic Seldon Function", generates a succession of "epochs" or "stages", s = 0, 1, 2, 3, 4, ..., in which a cumulative heterogeneous / non-amalgamative / non-reducing sum of all of these "ontological category" "qualifiers" increasingly and consecutively extends itself as the value of s rises:
[q/1]^(2^s) = q/1 + . . . + q/(2^s) .
KEY RULE: Repeated interpreted symbols in the denominators of the q numerators represent "revolutionary" possibility-irruptions of new, higher ontology / ontological categories, relative to that ontology which is represented by the category-symbol that is repeated in that denominator, in its single / non-repeated occurrence(s).
To take the capital-relation-incorporated "social relations of production" progression presented in Marx's Kapital as an example, suppose that we assign q/1 to q/C, or just C, standing for the human-social-relations-of-production-ontology category of "Commodities", as the beginning, or "arche' ", thesis".
Then the following dialectical model captures the "value-form" categorial progression of Marx's method of presentation as follows --
stage 0, s = 0, [C]^(2^0) = C^1 = C;
stage 1, s = 1, [C]^(2^1) = C^2 = q/C + q/CC = q/C + q/M = C + M,
such that M or q/M connotes the new social-relations-of-production-ontology category of "Monies", or the "de-abstraction"/"explicitization" of the "Money-relation" inherence in the "[industrial-]capital-[social-]relation[-of-production]", not yet extant in the more abstract, less- complex/less-concrete stage 0, but first-posited explicitly in stage 1;
stage 2, s = 2, [C]^(2^2) = C^4 = q/C + q/M + q/MC + q/MM = C + M + q/MC + K,
such that --
q/MC connotes "Money-Commodities" / conversion of Commodities into Monies / "real subsumption" of the Commodity barter-relation by the Money-relation / production of Commodities deliberately for the purpose of exchange for Money, rather than for barter / Money-mediated Circulation of Commodities, not yet extant in stages 0 and 1, but first posited explicitly in stage 2;
K or q/K connotes the social-relations-of-production-ontology-category of "Kapitals", not yet explicitly posited in stages 0 and 1, but first explicitly posited in stage 2, and signifying the "formal subsumption" of all previously-posited social-relations-of-production elements, or moments, by the Kapital-relation [and therefore connoting also the "antediluvian" species of capital -- usurers' capital; mercantile capital; latifundial, slavery-based agricultural production capital; merchant-orchestrated household-crafts manufacturing capital, etc., but not yet the "real subsumption by Capital" species of the Capital-relation -- industrial capital];
stage 3, s = 3, [C]^(2^3) = C^8 = C + M + q/MC + K + q/KC + q/KM + q/KMC + q/KK = C + M + q/MC + K + q/KC + q/KM + q/KMC + S,
such that --
q/KC = Commodity-Kapital / conversion of Commodities into Kapitals / "real subsumption" of the Commodity-relation by the Kapital-relation, not yet explicitly posited in stages 0 , 1, and 2, but first explicitly posited in stage 3;
q/KM = Money-Kapital / conversion of Monies into Kapitals / "real subsumption" of the Money-relation by the Kapital-relation, not yet explicitly posited in stages 0 , 1, and 2, but first explicitly posited in stage 3;
q/KMC = Conversion, by the Kapital-relation, of the total C-M-C'-... Money-mediated-Circulation of Commodities process into the process of the Circulation of Kapitals, not yet explicitly posited in stages 0 , 1, and 2, but first explicitly posited in stage 3, and signifying the completion of the "real subsumption" of all previously-irrupted social relations of production by the [fixed-capital-intensive, industrial-]Kapital-relation;
q/KK = q/S = S = the self-conversion / self-expropriation / self-fetter-bursting of the Kapital-relation; the explicit positing of the Association-relation of the Associated Producers / the revolutionary irruption of Socialism, as an inherent potential eventual consequence of the development of the [industrial] capital-relation, not explicitly posited in stages 0 , 1, and 2, but first explicitly posited in stage 3, and signifying the "formal subsumption" of all previously-irrupted social relations of production by the "Associated-Producers-relation", but not yet the "real subsumption" of those predecessor social relations of production by the "Associated-Producers-relation".
For those intrepid enough to want to check for themselves the above calculations, using the "Dyadic Seldon Function", the main rules ["axioms"] of the "Rules System" of the F.E.D. "First Dialectical Arithmetic", that govern these "purely qualitative", yet, nevertheless, "algorithmic", calculations, the wherewithal to do so can be found at --
http://www.dialectics.info/dialectics/Welcome.html
http://www.dialectics.info/dialectics/Correspondence.html
http://www.dialectics.info/dialectics/Correspondence_files/Letter17-06JUN2009.pdf
-- or at --
http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/Adventures-In-Dialectics-entry.htm
http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/DialogicLetters/DialogicLetters-entry.htm
http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/DialogicLetters/EightAxioms/EightAxioms.pdf
Regards,
Miguel
F.E.D. definitions of special terms applied above --
<<arche'>>
http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Arche/Arche.htm
[producers'] Association-relation
no definition is as yet available in the Clarifications Archive, but see --
http://www.equitism.org/Equitism/Theory/PoliticalEconomicDemocracy/PoliticalEconomicDemocracy.htm
<<aufheben>>
http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Aufheben/Aufheben.htm
[archeonic] consecuum
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Glossary_files/D.I.,Post,Phono-Neograms,%5BC%5D.w3_OCR.pdf
Capital-[social-]relation[-of-production] as human socio-ontological category
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Glossary_files/D.I.,Post,Phono-Neograms,%5BC%5D.w3_OCR.pdf
-- and pp. B-24 through B-33 in --
http://www.dialectics.info/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf
[barterable] Commodity-[social-]relation[-of-production] as human socio-ontological category
no definition is as yet available in the Clarifications Archive, but see p. B-29 in --
http://www.dialectics.info/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf
contra-thesis
no definition is as yet available in the Clarifications Archive, but see pp. 2 through 4 in --
http://www.dialectics.info/dialectics/Primer_files/8_Fract1-1_OCR.pdf
dialectical arithmetic
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Primer_files/8_Fract1-1_OCR.pdf
dialectical [meta-]model
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Primer_files/8_Fract1-1_OCR.pdf
http://www.dialectics.info/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf
meta-numerals
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf
Money-[social-]relation[-of-production] as human socio-ontological category
no definition is as yet available in the Clarifications Archive, but see pp. B-24 through B-33 in --
http://www.dialectics.info/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf
ontological qualifier
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf
ordinality
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf
purely-qualitative
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf
purely-quantitative
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf
[purely-]qualitative addition / sum
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf
[purely-]qualitative division
no definition is as yet available in the Clarifications Archive, but see --
http://www.dialectics.info/dialectics/Correspondence_files/Cover%20Letter%2014%2018MAY2008_OCR.pdf
[Dyadic, Triadic] Seldon Functions
http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/SeldonFunctions/SeldonFunctions.htm
[human-]social-relations-of-production [neo-]ontology
no definition is as yet available in the Clarifications Archive, but see pp. B-24 through B-33 in --
http://www.dialectics.info/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf
subsumption, formal, real
no definition is as yet available in the Clarifications Archive
<<arche'>>-thesis
no definition is as yet available in the Clarifications Archive, but see pp. 2 through 4 in --
http://www.dialectics.info/dialectics/Primer_files/8_Fract1-1_OCR.pdf
uni-thesis, partial, full
no definition is as yet available in the Clarifications Archive, but see pp. 2 through 4 in --
http://www.dialectics.info/dialectics/Primer_files/8_Fract1-1_OCR.pdf
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