Sunday, November 29, 2015

“Probability-Density Distributions-Dynamics” and Dialectical-Algebraic ‘Cumula’.

Probability-Density Distributions-Dynamics and Dialectical-Algebraic Cumula.

Dear Readers,

‘‘‘Solved/interpreted’’’ NQ_ meta-model cumula, up to, but not including, their ‘‘‘meristemal’’’, final terms, in cases of Dyadic Seldon Function meta-models’, can be grasped as qualitative partitionings -- as discrete, ‘‘‘ontological’’’ partitionings -- & as apt namings, of the content of their associatedprobability-density distributions”, or ‘‘‘expected frequency-of-encounter’’’ [with the given kind of «monad»] distributions.

These NQ_-algebrameta-modeled ‘‘‘frequency-of-encounter distributions’’’ are generally expected to be ‘‘‘hyperbolic’’’ distributions, like the one depicted below, for the F.E.D. dialectical theory of everything -- the ‘‘‘dialectic of nature’’’ -- meta-model, where every successive term/ categorogram represents an «arithmos» of «monads» with less zenith onto-mass than do its predecessor categorograms. 

Or, they are expected to be ‘‘‘exponential’’’ distributions, like those for many of the Seldonian psychohistorical dialectical meta-equations --

-- describing aspects of human nature/dynamics.  

E.g., for the case of the human-social formation(s) psychohistorical dialectical meta-equation, the zenith onto-mass for the “bands formation-category is estimated to have been far exceeded by the zenith onto-mass for the “camps formation-category, whose zenith onto-mass, in turn, is estimated to have been far less than the zenith onto-mass for the agricultural “villagesformation-category, and so on.

Either way, our expectations are for one-tailed distributions -- upper-tail-only [right-tail only] in cases of ‘‘‘hyperbolic’’’ distributions, lower-tail-only [left-tail-only] in cases of ‘‘‘exponential’’’ distributions.

Each stage/epoch/step advance, in such a meta-model meta-distribution, multi-distribution, or distribution of distributions, models a ‘‘‘meta-evolutionary’’’, ‘‘‘revolutionary’’’, ‘‘‘meta-dynamical’’’, ontology-changing/ontology-[net-]expanding event, and represents an expanded possibilities-cumulum, forming a new ontological frequency distribution, and ‘‘‘adding’’’ it into the cumulumof  the earlier frequency distributions, already extantized.

Such advance does so because it brings with it an increment of new terms, of new categorograms, that represent, in Dyadic Seldon Function cases, the first full development of the latest-irrupting frequency distribution, that was only introduced pre-vestigially in the immediately previous epoch/step, as well as bringing the harbinger of the next, new frequency distribution, a frequency distribution to be fully ‘‘‘populated’’’ & developed only in the next epoch/step.

In cases of Triadic Seldon Function NQ_-algebraic meta-model meta-distributions, each stage/epoch/step of advance completes the real subsumption extantization of the discrete partitions/categories representing the content of the distribution for the latest-irrupted new ontology, with a dialectical full synthesis, or full uni-category, culmination. 

The harbinger of the frequency distribution for the next-to-be-irrupted ontology begins only in the next epoch/step, in cases of Triadic Seldon Function NQ_-algebraic meta-model meta-distributions.    



Saturday, November 28, 2015

Mathematics Defined as 'Ideometry' -- The Deductive, Formal-Logical Moment of Its Meaning.

Mathematics Defined as 'Ideometry' -- The Deductive, Formal-Logical Moment of Its Meaning.

Dear Readers,

In the definition, given below, we do not define mathematics as ‘ideometry...’, only in some non-standard sense, that excludes the usual account of mathematics as “axioms” and “postulates” finding, as “rules of inference” and “primitives” deciding, and as “theorems proving”, from the former as the foundations of that proving activity.
The ‘ideometry’, the ‘measurement of the ideas’, that are the “primitives”, the “rules of inference”, and the “axioms/“postulates” of a [e.g., of a candidate, or in-development] axioms-system of mathematics, e.g., of a newly-developing branch, or application, of mathematics, is accomplished -- using the word “measured” in the expanded, F.E.D. sense -- by drawing out their [conjoint] deductive, formal-logical  consequences, and by the usefulness of those consequences -- consequences in the form of theorems, lemmas, and corollaries;  usefulness even if only in terms of the cognitive-esthetic pleasure of the mathematician/author in the system that (s)he has created, but, hopefully, also, usefulness in the less narcissistic sense of the scientific and technological, human-societal self-reproductive self-force contributions/benefits/utility consequences of that axioms-system, e.g., due to the capability of its equations to model/predict-for salient aspects of nature/human experience.

In the real human praxis of mathematics, as opposed to its mystified myths and fables,
axioms and postulates do not descend, or fall from the heavens, as immaculate conceptions, from above, from some transcendental realm.  Candidate axioms are tried-out, to see whether or not, and to what extent, their conjoint implications achieve the goals and motives to fulfill which the axioms-system is to be created in the first place. 

Axioms, postulates, primitives, and rules of inference, are not themselves accessible by deductive logic, as Plato pointed out, so long ago, but are the logical «arché» -- derived by dialectical means, according to Plato -- from which all else is to be deduced. 

In the process of the development of a new axiomatic system, the implications of the candidate «arché», in effect, feed back upon, and act back upon -- reflect/reflex upon, and reflux upon -- those candidates, via the active, living mediation of the human agent(s) of this development. 

This process results in changes to those candidate «arché» when their implications fall short of their desiderata, and, thus, also change the deductive consequences, in an iterative process, that continues as long as, e.g., the human agents of mathematics see need for, and hope for, the improvement of the axioms-system, relative to its desiderata.

Thus, the very development of a mathematical, axiomatic system is a self-reflexive function, and a self-refluxive function, conducted by human mathematicians, and is, in that sense, also a ‘‘‘nonlinear process’’’ -- a process of ‘ideometry’, mediated by formal deduction.



Friday, November 27, 2015

Seldon Speaks: ‘Intra-Duality’ & ‘Diachronicity’.

Seldon Speaks:  Intra-Duality & Diachronicity.

Dear Readers,

It is my pleasure to share with you, from time to time, the most effulgent excerpts from the scintillating sayings -- shared by him among we of F.E.D. -- of F.E.D.’s co-founder, Karl Seldon, such as the following [E.D. standard edits applied] --

Within every moment, within every eventity, among every local «arithmos» of eventities, intra-duality is at work, transforming the momentaneously apparently synchronic, into the diachronic, driving quantitative-into-qualitative, ontological change, hence driving time; constituting and reconstituting concrete duration with every increment of that [it]self-induced change[-of-[it]self].

-- of very recent vintage, a saying which I have christened Intra-Duality & Diachronicity.   



ANNOUNCEMENT: ‘The Dialectical Analysis of Arithmetics’, in 3 Volumes.

Dear Readers,

F.Y.I.:  The General Council of F.E.D. has, today, assigned, to me, the preparation of a 3-volume treatise, to be entitled The Dialectical Analysis of Arithmetics, and to be published, by the F.E.D. Press, during the years 2017-2019, for the purpose of elaborating our solutions to three of our major Seldonian meta-equations, which give their individual names as the individual titles for each of these three volumes --

Volume I.:      The Gödelian Dialectic of the Standard Arithmetics.
Volume II.:     The Dialectic of the Seldonian Dialectical Arithmetics.
Volume III.:   The Dialectic of the Arithmetics of Logic.

-- a task to which I look forward with great relish!   



Tuesday, November 24, 2015

'''Formal Subsumption''' and '''Real Subsumption''' in the Context of the Categorial-Dialectical Calculus.

Dear Readers,

Marx addresses a special case of «aufheben»-conservation/subsumption processes in the [unpublished] so-called “6th Chapter” of «Das Kapital», in the context of the self-development of the [incarnated and agented and reproduced by humans] capitals-system -- from an early ‘ascendance phase’ of that system, characterized by the ‘“[merely] formal subsumption of the labor-process under capital”’, toward a zenith, and then toward a ‘descendance phase’, of that system, increasingly characterized by the ‘“real subsumption of the labor process under capital”’.

In this blog entry, I focus on a more general case of dialectical, categorial subsumption, where -- and recurrently so -- a ‘‘‘formal subsumption’’’ of one or more priorly-irrupted or priorly-evoked ontological categor(y)(ies), by a most-recently-irrupted or most-recently-evoked ontological category, is succeeded by a ‘‘‘real subsumption’’’ of that/those earlier-irrupted or earlier-evoked ontological categories, by that most-recently-irrupted, or most-recently-evoked -- and therefore ‘‘‘meristemal’’’/ ‘‘‘vanguard’’’ -- ontological category.

An example of this progression from ‘‘‘formal’’’ to ‘‘‘real’’’ subsumption, in the context of systematic dialectics, is that for the simplest categorial-calculus dialectical meta-model of Marx’s «Das Kapital» --

for step 2, )-|-(2  =  () C ()2^2  =  C4  =   C  +  M  +  qMC   +  K;

for step 3, )-|-(3  =  () C ()2^3  =  C8  =  

C  +  M  +  qMC   +  K  +  qKC  + qKM  + qKMC + qKK.

-- wherein this meta-model exhibits, in its step 2, a ‘‘‘formal subsumption’’’ of the earlier-evoked categories of exchange-value/social relations of production -- Commodities and Monies -- by the ‘‘‘Kapitals’’’/wage-labors, exchange-value/social relation of production category. 

Formally, at step 2 of its presentation, its ‘‘‘Kapitals’’’ category already «aufheben»-subsumes its Commodities, Monies, and Money-mediated Circulations of Commodities [qMC] categories, by its mere, “additive” [‘+’], presence in this step, because K has become the new leading category, surpassing, hence subordinating and demoting, all previously-presented categories, in complexity, in thought-concreteness -- in determinateness -- and in aptness for the Domain [‘kapitalist society’] being presented, systematically, in «Das Kapital».

And yet, thereby, it has not yet presented the forms/categories which express the incorporation and integration of those three pre-evoked categories in[to] a self-reproducing system of [especially industrial] ‘‘‘Kapitals’’’. 

So far, in presentation step 2, K connotes mainly the still present descendants of the early, pre-capitalism, “antediluvian species of capital” -- primarily mercantile capital -- which are confined to the Monies/Commodities Circulations-processes of social reproduction, but which do not directly re-organize and dominate its ‘productions-processes’ as well.

It is in step 3 of this meta-model[ed] presentation that the ‘‘‘real subsumption’’’, by the ‘‘‘Kapitals’’’ category, of all of its earlier-evoked categories, arrives explicitly, in the form of the superposition [‘+’] of the categories ‘‘‘Commodity-Kapitals’’’ [qKC], ‘‘‘Money-Kapitals’’’ [qKM], and ‘‘‘Circulations of the Total Social Capital’’’ [qKMC].

The latter category, qKMC, connotes the [partial] take-over, subsumption, or appropriation, by the ‘‘‘Kapitals’’’ category, K or qK, of the earlier-evoked  category of the simple Circulations-of-Commodities-by-Monies processes [qMC], a take-over which evokes the existing category of the realization of surplus-value profits via the social circulations of ‘‘‘Commodity-Kapitals’’’ [qKC] and of ‘‘‘Money-Kapitals’’’ [qKM], i.e., of ‘‘‘Kapitals’’’ as a whole, in their alternating, “metamorphosizing” forms of ‘‘‘Commodity-Kapitals’’’ [qKC] and ‘‘‘Money-Kapitals’’’ [qKM], by which Kapital takes full command of both the circulations-processes and the productions-processes of human-societal self-reproduction.

An example of this progression from ‘‘‘formal’’’ to ‘‘‘real’’’ subsumption, in the context of [psycho]historical dialectics, is that for the hypothesis of Dr. Denise Schmandt-Besserat, regarding the first genesis of written language, in ancient Mesopotamia.

Early-on, per this hypothesis, temple dues contributions, as well as debts, were recorded via fired clay token ‘micro-icons’, that represented individual units of specific kinds of goods donated/owed, tokens which were kept together, late in the ‘meta-evolution’ of this accounting praxis, by depositing them into hollowed-out wet clay envelopes, envelopes that were later fired. 

Because these clay envelopes were opaque, auditing a debt, or a tithe, required breaking open the envelope containing the record thereof, and, thereafter, also required the labor of creating a new envelope to re-house the tokens representing that record. 

At some point, accountant-scribes began to press the hard, fired clay tokens, that were to be put inside a clay envelope, into the wet clay of the outer surface of that clay envelope, before depositing those clay tokens inside that clay envelope, and firing that envelope, thus leaving “2-dimensional” impressions of the tokens inside, on the outside of the envelope, thereby making auditing less laborious. 

Thus was born a new ‘meme of representation’ -- of 3-dimensional objects [e.g., the fired clay tokens] by “2-dimensional” marks in clay. 

For a long time, a dual, redundant representation system persisted -- token-marks on the outside of the clay envelopes, tokens full-blown inside those same clay envelopes. 

This dual, redundant representation phase represents the ‘‘‘formal subsumption’’’ phase, of the 3-D ‘tokenology’ meme, by the “2-D” ‘tokenography’ meme. 

However, at a certain critical point, scribe-accountants stopped hollowing out wet clay slabs, to form them into clay envelopes, and, instead, molded those wet clay slabs into solid wet-clay tablets, upon which “2-D” marks were made by impression, and by ‘“incision”’. 

This clay tablets stage instantiates the ‘‘‘real subsumption’’’ of the 3-D ‘micro-iconic’ representation meme, by the meme of “2-D” representation, and leads on to the full cuneiform system of writing.

In general, in NQ dialectical-algebraic terms, ‘‘‘formal subsumption’’’  is ‘sum-subsumption’, ‘additive [‘+’] subsumption’, or ‘superpositional subsumption’, by the content represented by the ‘‘‘meristemal’’’ category, for a given stage in, e.g., a Dyadic Seldon Function dialectical meta-model, of all of the earlier-extantized ontological-categorial content of that meta-model, via the qualitative superposition processes signified by the NQ versions of the “plus sign” -- non-amalgamative, momentary/stroboscopic non-interaction-operation separator symbols, that sign the momentaneous formation of the cumulum of ontologies present for that stage, just prior to the inception of the category-interactions/-multiplications that generate the successor stage/cumulum representation.

 In general, in NQ dialectical-algebraic terms, ‘‘‘real subsumption’’’  is ‘multiplicative subsumption’, or ‘product subsumption’, and ‘[allo-]hybridization subsumption’, by the content represented by the ‘‘‘meristemal’’’ category-symbol, for a given stage in, e.g., a Dyadic Seldon Function dialectical meta-model, when it generates its incremental symbols for the next-stage ontology representation, via that ‘‘‘meristemal’’’ category-symbol ‘ontologically-multiplying’ itself against/interacting with, the category-symbols representing all of the content of the earlier-extantized, and still-[possibly-]extant, ontological-categorial content of that meta-model that is represented in the given stage’s cumulum.