Wednesday, September 07, 2011

The F.E.D. 'Meta-Pythagorean Proposition'

[First of a Series]:

The F.E.D. 'Meta-Pythagorean Proposition' --

Dear Readers,

As I see it, the F.E.D. <<oeuvre>> rises upon the foundation of a series of immanent critiques of a seminal succession of previous mythopoeic, religious, philosophical, scientific, mathematical, and psychohistorical contributions to human civilization -- to "The Human Phenome".

I plan to summarize these immanent critiques here in a series of posts, mostly in the chronological order of the contributions critiqued, and stating with an immanent critique that F.E.D. explicitly posits, in module # 136 of their book A Dialectical "Theory of Everything", which is quoted in its entirety below, with commentary, by yours truly, interspersed.

It summarizes their immanent critique of the ideology of the ancient Pythagorean brotherhood-sisterhood, circa 570 to 500 B.C.E.

Note: F.E.D. describes themselves, in module # 231 of A Dialectical "Theory of Everything", as "a 'Meta-Pythagorean' sisterhood-brotherhood".

The F.E.D. immanent critique of the ideology of Pythagoreanism can be grasped heuristically, using the Q dialectical ideography, and using the step s = 1 Triadic Seldon Function, as follows --

)-|-(0     =


(Pythagoreanism)^(1)         =


)-|-(1   =



(Pythagoreanism))(Pythagoreanism)         =

((Pythagoreanism)(Pythagoreanism))    =

Pythagoreanism + Contra-Pythagoreanism + Trans-Pythagoreanism     =

P + C + q/CP     =

+ C + T


1 + q/2 + q/(2+1)

-- in which the "ideo-ontological category" P, or Pythagoreanism, functions as specific "ideo-<<arche'>>", or "thesis", and in which the "ideo-ontological category" C, or Contra-Pythagoreanism, functions as "contra-thesis", or immanent-/self-negation of P, or Pythagoreanism, and in which the "ideo-ontological category" q/CP, or Trans-Pythagoreanism, functions as "uni-thesis", i.e., as the reconciliation of, or "complex unity" of, Pythagoreanism and Contra-Pythagoreanism, or of C, and P.

The progression of propositions extracted below -- "Pythagorean" followed by "trans-Pythagorean", "critically-corrected" propositions -- from their module # 136, constitutes, to my reading, a kind of abbreviated Systematic-Dialectical presentation of the fruits of their immanent critique of Pythagoreanism, as if a succession of "syntheses", or of "uni-theses", with the usual Systematic-Dialectical gradient: from simple/abstract to more and more complex/thought-concrete, sequentially "explicitizing" the formerly implicit network of meanings inherent in the Pythagorean <<arche'>> --

"All is <<arithmos>>".

-- or --

'''<<Arithmoi>> constitute the <<Kosmos>>'''.

-- usually, and misleadingly, translated into modern English as --

"All is number".

-- or as --

'''Numbers constitute the cosmos'''.

F.E.D. holds that <<arithmos>>, in ancient Greek, meant "assemblage of qualitative <<monads>>", i.e., "number of qualitative units".

This is not at all the same meaning as that of the more abstract and purely-quantitative meaning of, e.g., the English word "number", ever since the end of the Dark Ages that separate the "modern" human phenome from the ancient Mediterranean one.

Realizing this "gulf of meaning-difference" helps us to hear the ancient Pythagoreans' <<arche'>> claim in a less mystical, more concrete way --

'''Populations of individual units of various kinds constitute the cosmos'''.

-- that is, "things", "units", do not come as absolute "singletons", but arise in "multiplicities of similars", forming various "kinds" or <<gene>>.

This realization helps us to no longer see so much mystery as to why, to our ancient Mediterranean human-phenomic progenitors, "one" was not [a] "number" -- not an <<arithmos>>; not "quantitative" -- but was just a single, qualitative <<monad>>; why, for them, the domain of "number", of "the quantitative", began with two, not with one, and why "zero" was not an <<arithmos>>-concept for them.

It helps us to understand the "mysteries" cited by the following authoritative, "definitionary" source on the word <<arithmos>> --

"arithmos: number; arithmêtikê; the science of number. Zero was unknown as a number and one also was not counted as a number, the first number being the duas – two. From the Pythagoreans, ton arithmon nomizontes arkhên einai – who consider number to be the first principle (Ar. Met. 986a15) – number played a great part in metaphysics, especially in Plato’s unwritten doctrines, involving obscure distinctions of e.g. sumblêtoi and asumblêtoiaddible and non-addible number." [J. O. Urmson, The Greek Philosophical Vocabulary, Gerald Duckworth & Co., Ltd. [London: 1990], pp. 31-32, [emphasis added]].

This veritable contradiction between ancient and modern Occidental theories of number, and of arithmetic, were glaringly evident in perhaps the last great work of ancient science to survive, the work that laid the foundation for modern algebra, and, in many ways, for modern ideographical mathematics entire -- the <<Arithmetike'>>, by Diophantus of Alexandria.

In that work, the "abstraction of number" from ordinary-language discourse was made in a crucially-different way from the way in which we and our ancestors have made it ever since the Dark Ages gave way to the present "Dim Ages".

From "two apples plus two apples makes four apples"., we moderns abstract only the generic quantifiers:

2 + 2 = 4

Diophantus abstracted, in generic form, the ontological ["kind-of-thing"] qualifiers as well as the ontological quantifiers.

He abstracted a generic <<Mo-nad>>[-ic qualifier], denoted by the "syncopated" abbreviation "letter-omicron-atop-letter-Mu", or M^o, as well as its quantifiers, also using, by way of "gematria", the primed second Greek letter, "beta-prime", to denote what we take to be "2", and the primed fourth Greek letter, "delta-prime", to denote what we take to be "4":

2M^o + 2M^o   =   4M^o,

or, with considerably greater faithfulness -- within standard typographical limitations -- to the actual Diophantine "syncopated-algebraic" handwritten notation extant in the portions of Diophantus's manuscripts that survived the last Dark Ages:

M^o b'b' <<aequalis>> M^o d'.

Diophantus's mathematical, arithmetical <<monad>>, or unit, M^o, is, in fact, a forerunner, or "pre-vestige", of F.E.D.'s qualitative dialectical-arithmetical units, like q/n [for any n in the space of "Natural" Numbers, N].

More exactingly, with the two English characters, Mu, denoting the single-character 12th Greek letter <<mu>>, Diophantus's M^o is a "pre-vestige" of the value Mu/(u/0), or Mu-subscript-(u-subscript-0), Muu0, in F.E.D.'s second full uni-system dialectical-arithmetical axiomatic system, the subscript-wq/7 (---] "subscript-WMu_" axiomatic system of dialectical arithmetic, which "internalizes" the "subscript-WU_" first full uni-system dialectical-arithmetical axioms-system, both being based upon the zero-including axioms-system of the "Standard" arithmetic of the "Whole Numbers", W, whose space is W = { 0, 1, 2, 3, 4, 5, 6, 7 . . . }, and wherein Mu/(u/0) functions as a "dimension-less" unit, or %-like unit, similar to M^o, e.g., --

1% + 1%  =   2%

a'M^o + a'M^o   =   b'M^o

1Muu0 + 1Muu0   =   (1 + 1)Muu0   =   2Muu0

-- or --

1Mu/(u/0) + 1Mu/(u/0)   =   (1 + 1)Mu/(u/0)   =   2Mu/(u/0)

-- and --

M^o x a'M^o   =   a'M^o

1Mu/(u/0) x 1Mu/(u/0)   =   1Mu/( u/0 + u/0 )   =   Mu/(u/0)

-- or --

1Muu0 x 1Muu0   =   1Muu0+u0   =   Muu0

The marked discrepancy between the ancient and the modern Occidental meanings, and scientific theories, of the concept of "number" / <<arithmos>> represents a major fount and foundation for the F.E.D. "meta-science" of '''psychohistory''', and for their "psychohistorical dialectics", as well as for their Marxian paradigm as a whole, which they name "Psychohistorical Materialism ".

F.E.D. accounts for this discrepancy in "number" theory -- in the concept of the "quantitative", and in the intrinsically-related concept of the "qualitative" -- in terms of the less-permeating molding of the ancients experience by their emerging, "protoic" law of value, and in terms of the ever-more-pervasively-permeating, unconscious molding of the modern <<mentalite'>>, by the Capital-relation's law of value.

Marx's "elementary value form" -- the single, <<arche'>> socio-ontological category from which all of the rest of the categorial progression of Capital dialectically flows -- shows how, in our normal ignorance of "the socially-necessary abstract human labor-time presently required to reproduce a commodity" as core determinant of a commodity's prevailing monetary value/price central tendency, and, therefore, in our ignorance of the core, qualitative meaning of the monetary unit -- e.g., of $ -- our incessant commodity purchase [and sale] activities in our modern = capital-centered society unconsciously inculcate in us a fallacious sense of the purely-quantitative exchange-equating of qualitatively heterogeneous commodities, thus continually reinforcing a mentality of the abstraction of pure quantity, of which money-value quantities become the deeply-addicting paradigm, permeating the collective, "social-unconscious mind" of humanity -- the unconscious ideology-forming domain of the human phenome -- to its deepest reaches, and unconsciously deforming even the most conscious summits of the capital epoch's ideological sciences with a profound fetishism of heterogeneity-denying, "qualitativity"-denying "quantism".

F.E.D. wrote, in their module #136 --

1. «Arithmoi» constitute the «Kosmos».

[To hold that <<arithmoi>> of <<monads>> constitute the <<kosmos>> is, as already pointed out above, merely to emphasize a commonplace empirical observation: "individuals" of all kinds come in populations -- M.D.].

2. «Arithmoi» constitute the «Kosmos», via [self-]«aufheben» meta-monadization.

[In this second step, F.E.D. adds their noticing of the process of '[self-]-meta-unit-ization', breaking with the tendencies toward a Parmenidean stasis view of the <<kosmos>>, by naming/describing the mechanism [or '''organism'''] in the universe that generates new <<arithmoi>>, new "numbers", new "kinds-of-being"/<<gene>>:  new ontology. F.E.D. also, in the process, notes that this generator is an <<aufheben>> one, i.e., a dialectical, one -- M.D.].

3. «Arithmoi» constitute the «Kosmos», via self-reflexive, nonlinear,self-iterated [self-]«aufheben» [self-]meta-monadization.

[In this step three, F.E.D. adds notice of the interconnexions that they see, among '''dialecticity''', "self-reflexivity", and mathematical, differential equation "nonlinearity" -- M.D.].

4. «Arithmoi» constitute the «Kosmos», via tau-Epochal, self-reflexive, self-refluxive, ‘‘‘karmic’’’, nonlinear,self-iterated [self-]«aufheben» [self-]meta-monadization.

[In this fourth step, F.E.D. adds notice of the interconnexions that they see, among '''dialecticity''', and "self-refluxivity", as '''karmicity''', the latter as a fruition, I think, of their immanent critique of ancient Oriental religious and mystical ideologies. The self-iterated "epochality" of the posited "self-meta-<<monad>>-izations" is also explicitly noted here for the first time in this sequence of propositions. "Self-iterated" as used in above means essentially "suitable to be modeled, mathematically, via a "self-reflexive function", such as one of the "Seldon Functions" -- M.D.].

5. «Arithmoi» constitute the «Kosmos», via tau-Epochal, self-reflexive, self-refluxive, ‘‘‘karmic’’’, nonlinear, self-iterated [self-]«aufheben» [self-]meta-monadization, producing natural-historical time as an accelerating progression of «aufheben», or ‘‘‘evolute’’’, evolutions, each characterized by a quantitatively expanding self-reproduction of the then-leadingArithmic, Monadic population, which, upon reaching critical population densities, punctuate those natural evolutions with ‘‘‘meta-evolutions’’’ -- self-revolutions in Nature -- manifesting as meta-finite, self-bifurcation, ‘‘‘resonant’’’, meta-monadizing singularities, superseding each current leading «Arithmos» with a new one, of new qualo-fractal scale/level; a new Quality/Kind/«Genos», of universal, Arithmic’, ‘[Meta-]Monadic ontology.

[In this fifth/final step for this module, F.E.D. irrupts a whole cascade of "explicitization" of former "implicitudes" of their "Trans-Pythagorean" account of the <<kosmos>>.

They connect "quantitative change" -- quantitatively-expanding self-reproduction and "[self-]densification" of monadic populations -- with "qualitative change" -- with qualitatively-expanding self-reproduction of monadic populations in the form of the "irruption" of new, meta-monadic populations -- new "kinds", new <<gene>>, new ontology -- which they identify as the fruits of "self-meta-evolutions", or "self-revolutions", in Nature, and which they also connect with the construction, by Nature, of new "qualo-fractal" scales/levels of being, and with "singularities" in Nature, which they further characterize as "meta-finite", and "resonant", and as "self-bifurcations".

They thus relate their new paradigm, in an "immanently critical" fashion -- to "dynamical systems theory", or to so-called "chaos theory", as the theory of [especially nonlinear, and therefore singularity-prone] "total" differential equations-systems, or "ordinary" differential equations-systems, as applied to the modeling of, e.g., "natural", "physical" dynamical systems [systems that are typically, for F.E.D., both "self-dynamical", or "self-dynamizing", as well as "other-systems-dynamized"], using the language of "total differential equations", i.e., of Real numbers or Complex numbers algebra, supplemented, or "extended", to incorporate two additional, mutually-inverse, '''[hyper-]arithmetical''' operations, operations #7 and #8, in addition to the traditional, paired, six: addition/subtraction, multiplication/division, raising to the rth power/taking the rth root -- namely integration/differentiation, and using only "total", as opposed to "partial", differentiation operations.

They also critique the modern, abstract concept of time, by re-defining time in terms of its continual [re-]production process in the concrete processes of universal evolution and, especially, of universal "meta-evolution", or of "universal self-revolution" -- M.D.].



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