Saturday, May 04, 2013

The Dialectic of Ancient Philosophy, rendered in the "local" typography

Dear Readers,

A new, second version of F.E.D. Vignette #12, was posted, earlier today, to the website.  

The URLs for that version are the following --

-- and I have also rendered this second version, below, using the typography local to this blog.




 F.E.D. Vignette #12 --

The [Psycho]Historical Dialectic of
 Ancient Mediterranean Philosophy
through Platos Philosophical Developments

 by Aoristos Dyosphainthos

Authors Preface.  The purpose of F.E.D. Vignette #12 is to present an abbreviated E.D. Dialectical Meta-Model Meta-Equation for the [psycho]history -- and for the [psycho]historical dialectic -- of the progression of Ancient Mediterranean philosophies through the two philosophies espoused by Plato, his “early” and his “final” philosophies.

This Dialectical Meta-Model Meta-Equation can also serve as a worked, “cook-book” example, and sample, of the application of the NQ dialectical algebra to summarize interconnections of, and to “solve for”, the psychohistorical-dialecticalideo-meta-dynamicsof key episodes in the [psycho]historical progress of The Human Phenome.

A Note about the On-Line Availability of Definitions of F.E.D. Key Technical Terms.  Definitions of Encyclopedia Dialectica technical terms, including E.D. ‘neologia’, are available on-line via the following URLs --

-- by clicking on the links associated with each such term, listed, alphabetically, on the web-pages of the links above.

Definitions of the Encyclopedia Dialectica special terms most fundamental to this vignette are linked-to below --


«arithmos» and «arithmoi»


Historical Dialectics

NQ dialectical arithmetic/algebra

Psychohistorical Dialectics

-- and we plan to expand these definitions resources as the Encyclopedia Dialectica  Dictionary Project unfolds.

[Note:  ‘‘‘Arithmetical Quantifiers’’’ vs. ‘Arithmetical Qualifiers’.  In the phrase “3 apples”, we term “3” the “arithmetical [“pure”-]quantifier”, and “apples” the ‘‘‘ontological’’’ -- or kind of thing -- ‘‘‘qualifier’’’.  In the phrase “3 pounds of apples”, we term “pounds” the metrical[-unit] qualifier’ -- or ‘‘‘unit of measure qualifier’’’ -- quantified by the 3, which, together, quanto-qualify the ontological qualifier’, “apples”.  A key use-value of the dialectical arithmetics is to provide algorithmic, ideographical-symbolic systems for the various kinds of ‘arithmetical qualifiers’, both with and without the co-presence of ‘‘‘arithmetical quantifiers’’’.].

I.  Introduction to the Psychohistorical Dialectic of Ancient Mediterranean Philosophy.  The historical expanse of “Ancient Mediterranean [“Occidental”] Philosophy”, up to and including both of the two major stages of the philosophies of Plato, offers a stunningly rich diversity of ‘human memetic content’ -- of psychohistorical material.

The task of framing a unified model, let alone a dialectical-mathematical meta-model, to encompass all or most of the history of this vast human-phenomic proliferation, as an ideo-meta-genealogy, all following from a single philosophy as «arché», or starting point, via the iterated self-reflexion, and self-refluxion, of that «arché», and of the ideo-cumula spawned by that self-iteration, would seem to be a daunting assignment, to say the least.

Many difficult choices face the ‘‘‘psychohistorian’’’, the dialectical meta-modeler who pursues such a task, not least of which is the choice of «arché» philosophy, of the ultimate ideo-ancestor, of all that follows it in the psychoalgebraic representations of the meta-model-generatedprogression of multi-category [multi-philosophy] ideo-cumula.

One might want to consider, as candidates for this «arché» philosophy -- i.e., for the ‘‘‘kernel’’’ or ‘‘‘seed’’’ of the meta-model Seldon Function -- the early philosophies of the «arché» of the «kosmos» -- e.g., the philosophy of Anaximander [moisture as «arché»], the philosophy of Anaximenes [air/breath-of-life as «arché»], and the philosophy of Pythagoras [the origin of the number sequence as «arché»], etc.

For the purposes of the guided meta-model construction exercise exposited in this vignette, we have chosen the philosophy of Herakleitos as «arché», and have sought thereby to encompass, by the connotative entailments of its ‘‘‘Triadic Seldon Function’’’ self-iterata, and via two successive such self-iterations, or ‘‘‘negations of negations’’’, of our Herakleitean «arché», both the philosophy of Parmenides, and, then, also Plato’s early philosophy of the supremacy of The Forms, and, finally, Plato’s final recorded philosophy, of the supremacy of «autokinesis».

The dialectical-algebraic language that we will employ in constructing this dialectical meta-model is that of the NQ dialectical arithmetic, the first and simplest explicitly dialectical arithmetic in the dialectical progression of dialectical arithmetics discovered by our co-founder, Karl Seldon.

II.  E.D. Standard Interpretations for the Initial Generic Ordinal Qualifiers of the NQ Arithmetic.  The first four, first-order-logic, Dedekind-Peano Postulates for the “Natural” Numbers focus on ordinality, not cardinality, viz. --

1.  1 is a “Natural Number”.

2.  The successor of any “Natural Number” is also a “Natural Number”.

3.  No two, distinct “Natural Numbers” have the same successor.   

4.  1 is not the successor of any “Natural Number”, i.e., 1 has no ancestor within the “Natural Numbers”.

-- defining the essence of the “Natural Numbers” explicitly in terms of [apparently purely-]quantitative ordinality.  

In keeping with this focus on the ordinal, Seldon defines the system of the NQ dialectical arithmetic -- the first antithesis-system, or contra-system, to the “Natural Numbers” system -- in terms of qualitative ordinality.   

The NQ , which he also calls the meta-Natural meta-Numbers, are, in their simplest, least-interpreted essence, a consecutive sequence of ‘meta-numeral’ ideograms representing the successive qualities, not quantities, of ordinality -- the quality of first-ness’, followed by the quality of second-ness’, followed by the quality of third-ness’, and so on... -- satisfying the four first-order-logic contra-Peanic, Qualo-Peanic axioms:

1'.  The ordinal qualifier for the quality of first-ness’ is an element of the consecuum of generic ordinal qualifiers.

2'.  The successor of any element of the consecuum of generic ordinal qualifiers is also an element of same.

3'.  Any two, distinct ordinal qualifiers have qualitatively unequal successors.   

4'.  The ordinal qualifier for the quality of first-ness’ is «arché»:  not the successor of any element of its consecuum.

The symbols, or meta-numerals, that stand for the meta-numbersof the NQ archeonic consecuum are derived, syntactically, in a way which represents the semantic self-subsumption, self-subordination, or self-demotion [dialectical, self-«aufheben» self-negation] of the “Natural Numbers”.   

That process is the positive fruition of the dialectical, immanent self-critique of the “Natural Numbers”, which divulges the NQ as the implicit, most extreme known opposite, Non-Standard Model” of the “Standard”, Peano “Natural Numbers”.   

This involves the turning of generic ordinal quantifiers of the “Natural Numbers” into the generic ordinal qualifiers of the NQ meta-Natural meta-Numbers.   

The conceptual self-subsumptionof the quantitative ordinality intended by the Dedekind-Peano Postulates surfaces the NQ as their hitherto hidden, implicit intra-dual, based upon the generic quality of ordinality, a ‘‘‘genericity’’’ which we represent by the meta-numeralic ideogram ‘q’.   

That meta-numeral component represents qualitative ordinality’, or ordinal quality, in general:  just ‘q’, or, more fully expressed, just qN.

To fully express, meta-numeral-y, or ideographically, the consecuum of specific ordinal qualities, namely --

NQ  =  {first-ness’; second-ness’; third-ness’, etc.}

-- we must add a second meta-numeral component, via ‘‘‘subordinating’’’ specificNatural Numbers”, as specific ordinal quantifiers, to the generic ordinal qualifier symbol ‘q’, by subscripting those specificNatural Numbers” to a ‘script-level’ ‘q’ above them, yielding --

NQ  =  { q1, q2, q3, ... } [in which each meta-number is an «arithmos eidetikos»],   

vs. N  =  { 1, 2, 3, ... }. 

Note that this opposition of an arithmetical system of purely-quantitative ordinality, based upon the N, versus an arithmetical system of purely-qualitative ordinality, based on the NQ, is not a radical dualism, imagined as an absolute, irreconcilable diremption between an absolute quantitative and an absolute qualitative.   

This opposition is, on the contrary, a dialectical antithesis-sum.   

The N quantifiers are still there -- still present -- in, or ‘‘‘under’’’, the NQ qualifiers, though subsumed, subordinated, demoted -- as their subscripts or denominators:  The N quantifiers are still contained in[side] the NQ qualifiers.   

That is, NQ qualifiers are «aufheben» determinate negations / conservations / elevations of N quantifiers.

For this first layer of interpretation of these purely-qualitative NQ meta-numbers -- which does not yet make explicit their universal interpretability for the modeling of dialectical progressions -- this is all that they represent:  abstract temporality; [abstract chronological] order; generic ordered-ness; the consecutive succession of qualitative ordinality; the consecuum of order quality or of order qualities.

But even here, at this minimally-interpreted stage of the construction of the NQ dialectical arithmetic, there is already a kind of generic connotative entailment at work. 

True, it is but a shadow, and but a ‘pre-vestigial’ harbinger, of the richness of the kind of particularity of categorial followership that drives forward, intuitively, the dialectical, purely-qualitative logic of the more concrete, more specific dialectical-algebraic interpretations thereof.   

A case in point is exemplified in the very meta-model of the dialectic of the Ancient Mediterranean Philosophies, constructed herein.

This generic connotative entailmentcan be formulated as follows:  second-ness’ follows -- and even follows from --first-ness’; third-ness’ follows [from] second-ness’, and so on.

In the next section, the construction, by iterated interpretations layering, of the Seldonian first dialectical arithmetic will advance from this harbinger of connotative entailment to the following, still generic, but at last also explicitly dialectical, form of connotative entailment ordinality:  first full antithesis follows from the self-interaction of [«arché»-]thesis; first full synthesis follows from the mutual interaction of first full thesis and first full antithesis, and so on.

III.  Triadic Seldon Function Interpretation of the Initial Generic NQ Ontological Category Qualifiers. 
The generic form for the functions-family of the Seldon Functions is that of a generic cumulum symbol [ |-|-| ] on the LHS [Left-Hand Side] of  the dialectical meta-equation, equated to an RHS expression representing self-reflexive operation of an [«arché», ‘‘‘seed’’’, ‘‘‘cell-form’’’, or ultimate ancestor ontological category symbol [ q1 ] -- indicating its recurring self-reflexionvia a ‘meta-exponentiated’, monotonically increasing whole-number-valued ‘‘‘independent variable’’’ [ h ] -- on the RHS of the generic Seldon Function equation, viz. [with ‘generic-ness’ connoted by the “rectangular” motif of the symbols-set]:

|-|-|h      =      [ q1 ]vh, for h in { 0, 1, 2, 3, . . . }.                           

If v = 2, the Generic Seldon Function above is said to belong to the Dyadic Seldon Function sub-family. 
If v = 3, the Generic Seldon Function above is said to belong to the Triadic Seldon Function sub-family.   

Our remarks herein are concentrated on the Triadic Seldon Functions, as the dialectical-mathematical meta-equation’, modeling the dialectical progression of the Ancient Mediterranean Philosophies, exposited herein, is of the v = 3 variety.

With v = 3, and selecting that special generic Triadic Seldon Function form that we reserve for an historical dialectic, or for a psychohistorical dialectic, the form of the meta-model meta-equation to be constructed herein becomes, more specifically --
>-|-<t­         =        < qa >3t ­,

for t in { 0, 1, 2, 3, . . . }.                       

-- wherein the symbol t, replacing the more generic symbol h, takes on ‘temporal’ connotations, representing successive historical periods, or “epochs”, and wherein, in general, the “angular” motif of the entire symbols-set used is to connote the [psycho]historical domain of dialectical meta-modeling.
The Seldon Functions bring with them a further, second layer of interpretation of the NQ qualifiers,  

{ q1, q2, q3, ... }

by which they are interpreted as qualifiers that symbolize generic dialectical ontological categories, e.g., as ‘‘‘thesis’’’ categories, or as full or partial contra-thesis categories, or as full or partial uni-thesis categories.

If we assign [ [---> ] the «arché»-thesis category, qa, to the generic NQ qualifier meta-number, q1, as signed byqa [---> q1’, and if we can discern that qa, and all of its successor-categories, and their cumula, as generated by its successive, cumulative, ‘Seldon-functional self-operations’, connote «aufheben» operators, that is, dialectical negation operators, then the Triadic Seldon Function is seen to signify, under the axioms of the system of arithmetic of the NQ meta-numbers[ ], a self-iterated, cumulative recurrence of dialecticalnegations of [the] negations.

With every [unit] increase in t, the Triadic Seldon Function ‘formulaic recipe’ calls for the triadic self-operation of the result of the previous triadic self-operation, i.e., for a negation of the negation of the result of the previous negation of the negation.   

Only for t = 0 -- only for the case in which no self-operation occurs -- is the “result” a singleton [ideo-] ontological category symbol, the symbol for the «arché» [ideo-]ontological category alone, instead of that “result” taking the form of a cumulum of three or more such symbols, i.e., a “non-amalgamative sum” [cf. Musès], or «a-sumbletoi» sum [cf. Plato], of ‘[ideo-]ontological category’ symbols, since 30 =  1:

|-|-|0   =    [ qa ]30       =         [ qa ]1      =       [ qa]      =     qa.

For example, if we take epoch t = 1, and denote the «arché» ontological category simply by a, for syntactical convenience, then the Triadic Seldon Function calls for the following, as per the NQ axioms, because 31 =  3 --

 |-|-|1    =    [ a ]31      =       [ a ]3       =        [ a ] x [ a ] x [ a ]     = 

a  +  b  +  c.

-- wherein a [---> q1 denotes the «arché» category or ‘‘‘thesis’’’, b [---> q2 the first contra-category or ‘‘‘antithesis’’’, and c [---> q3 the first uni-categoryor ‘‘‘synthesis’’’, with ‘+’ standing for a generalized addition operation, that covers the addition of qualitatively distinct terms, and with ‘x’ standing for a generalized multiplication operation, that covers multiplication of NQ qualifiers.

If, now interpreting a as the «arché» of a [psycho]historical dialectic, we take 

< a > x < a > x < a >’ 

as connoting the associative grouping 

< <a > x < < a > x < a > > >’, 


< a > 

denoting the category to be dialectically negated, and then dialectically negated again, and with 

< < a > x < a > >’ 

denoting the first negation of that category, and with 

< < a > x < < a >2 > >’ 

as the next, second negation, then we have

< < a > x < < a > x < a > > >’, 

as a whole, connoting the first dialectical negation of the negation, yielding the

first triad of ‘‘‘thesis + antithesis + synthesis’’’, viz. --

< < a > x < a > >     =     qa   x   qaa     =     qa   x   qb       =       a  +  b  

-- and --

< a > x < a > x < a >     =    < a > x < < a > x < a > >    =     

< a > x < a  +  b >    =     

a  +  b  +  qba    =    a  +  b  +  c

-- which, in terms of the generic, minimally-interpreted NQ arithmetics, is a dialectical interpretation of the generic --

|-|-|1     =       [ q1 ]31     =      [ q1]3     =     [ q1 ] x [ q1 ] x [ q1 ]     =     

[ q1 ] x [  [ q1 ] x [ q1 ] ]    =     

[ q1 ] x [ q1 + q1+1 ]    =    [ q1 ] x [ q1 +  q2 ]    =    

[ [ [ q1 ] x [ q1 ] ] + [ [ q1 ] x [ q2 ] ] ]     =    

[ [ q1 +  q1+1 ] +  [ q2 +  q1+2 ] ]        q1  +  q2  +  q3

given that   

q2 + q2   =   q2.

The second iteration of this dialectical negation of the negation, corresponding to the consecutively next value of t, namely, t = 2, yields the following, ontologically-expanded cumulum of ontological categories -- a triad of triads, consisting of 9 consecutive ontological categories:

|-|-|2   =   [ a ]32 =   [ a ]9    =    [    [ a ]3 ]3   =   [ a  +  b  +  c ]3           

=    [ a  +  b  +  c ] x [ a  +  b  +  c ] x [ a  +  b  +  c ]

=    [ a  +  b  +  c ] x [ [ a  +  b  +  c ] x [ a  +  b  +  c ] ]     

=     a  +  b  +  c  +  d  +  e  +  f  +  g  +  h  +  i.

The additional 6 ontological category-symbols above are dialectically interpreted, per the E.D. standard, as follows, in terms of their generic dialectical interpretation:

d  =  fourth ontological category, first partial contra-category;
e  =  fifth ontological category, second partial contra-category;   
f   =  sixth ontological category, second full contra-category;   
g  =  seventh ontological category, first partial uni-category;   
h  =  eighth ontological category, second partial uni-category;   
i  =  ninth ontological category, second full uni-category.

We will not here pursue this E.D. standard dialectical interpretation of the ontological categories generated by the generic Triadic Seldon Function beyond t = 2, because the ‘‘‘solution’’’ -- or semantification -- of the category-terms generated by the dialectical-mathematical meta-model meta-equationconstructed in this vignette [whose terms are generated, initially, as algebraic unknowns, terms of unknown meaning], as presented herein, does not extend beyond that second self-iteration for that meta-model.

The purely-qualitative calculations illustrated above describe our expectations for this meta-modelin terms of generic characterizations of the successive, consecutive dialectical categories. 

The next section addresses the heart of this meta-model -- the specific meanings of the generic dialectical categories as applied to the special case of the psychohistorical dialectic of the Ancient Mediterranean Philosophies.

IV.  E.D. Solution for the Meta-Model Meta-Equation of our Dialectic of Ancient Philosophy.  We have selected, as the «arché» of the universe of discourse of Ancient Mediterranean Philosophy, the philosophy of Herakleitos [circa 540-475 B.C.E.], and we denote that philosophy, in the meta-model thereof, constructed in this section, by the symbol H, or qH, [---> q1.

The few fragments of Herakleitos’s writings that survived the last Dark Ages indicate that he held a view of reality as a flowing continuum/universal flux, characterized by constant change, a uni-category of constancy &/vs. change:

Everything changes and nothing remains still ... and ... you cannot step twice into the same stream.;

We both step and do not step in the same rivers. We are and are not. ;

All things are an interchange for fire, and fire for all things, just like goods for gold and gold for goods.;

We must know that war is common to all and strife is justice, and that all things come into being through strife necessarily.;

[Diogenes Laërtius interpreting]: All things come into being by conflict of opposites, and the sum of things flows like a stream.;

There is a harmony in the bending back as in the case of the bow and the lyre.[Emphasis added by A.D.]

Fire serves as metaphor, in Herakleitos’s philosophy, for the primal substance, or «arché», of the «kosmos»; the ultimate origin of all other things.

Our next step, then, is to dialectically negate this H, using H itself as the dialectical negation operation for itself --

< < H >  x  < H > >     =     qH  +  qHH     =     qH  +  q?       

=      H  +  ?  

-- and to inquire, then, as to what might be an apt historical, Ancient Mediterranean meaning, for the algorithmically-generated new symbol  


as to what might have been the new, historical, philosophical, ideo-ontological category; the new kind of philosophical ideas, of philosophical psychohistorical material -- that irrupted from out of this immanent, self-confrontation -- this self-critique -- of H or qH, by H or qH. 

I.e., our next task is to solve for” qHH.

What we find, as our best candidate for the historical/philosophical meaning of our ‘‘‘algebraic unknown’’’,


is the later emerged contra-philosophy -- contra-Herakleitean philosophy -- of Parmenides, denoted herein by P or qP.

The philosophy of Parmenides [515-450 B.C.E.], as determined from surviving accounts of his views, is one of a claimed eternal changelessness of reality -- a “reality” in which change is only a human delusion:

There is one story left, one road:  that it is.  And on this road there are very many signs that being is uncreated and imperishable, whole, unique, unwavering, and complete.[Emphasis added by A.D.]

Parmenideanism is a kind of absolutist meta-monad-ization of Heraclitean diversity and flux:  a single «monad» of being posited not as the «aufheben» of the vast multiplicity of «monads»  of diversity/flux, but claimed to be their absolute obliteration, as falsity, in a singular truth of absolute being.

Our solution to the  

< < H > x  < H > >  

part of  

H3   =   < < H > x  < < H > x  < H > > >  

is thus

 qHH   =   qP   =    

P, [---> q2.   

But what of the rest of 

 < < H > x  < < H > x  < H > > >

what of
< < H > x < qH  +  qHH  > >   =    H x < H + P >?   

What historic meme of the Ancient Occidental Human Phenome corresponds to  qPH, [---> q3

What we find, as our best candidate for the historical/philosophical meaning of our ‘‘‘algebraic unknown’’’,  


is the later emerged uni-philosophy of Plato, the philosophy of the transcendental «Eide», also denoted herein by  

E or qE.

The earlier philosophy of Plato, as determined from his extensive written remains, is one combining the opposing philosophies of Parmenides and Herakleitos into a Parmenidean-dominant complex unity [cf. Hegel], or ‘‘‘dialectical synthesis’’’, positing a ruling, transcendental, eternally changeless and “true” reality -- the reality of the «Eide» or «Idea»  -- above, and controlling, a subordinate, truth-falsifying, illusional realm of human sensuous experience below.

This Platonian complex unity,  


is not a simple welding-together of a Parmenidean realm, P, “atop” a Herakleitean one, H.   

Their unifying complex features a middle realm, mediating and bridging the Parmenidean realm above this ‘‘‘median’’’, and/with the Herakleitean realm below that ‘‘‘median’’’.

Magisterial Plato scholar Jacob Klein describes this Platonian unifying complex in the following terms --

...While the numbersarithmoi»; assemblages of unitsA.D.] with which the arithmetician deals, the arithmoi mathematikoi or monadikoi [abstract, generic, idealized, qualitatively-identical, homogeneous “monads” or [ideal[ized], abstract qualitative unitsA.D.] are capable of being counted up, i.e., added, so that, for instance, eight monads [eight ‘ideo-monads; eight abstract ideal[ized]-units, unities, or idea-a-tomsA.D.] and ten monads make precisely eighteen monads together, the assemblages of eide [of ‘mental seeings or mental visions; of ultimate ancestor «ideas»A.D.], the arithmoi eidetikoi [assemblages, ensembles, ‘‘‘sets’’’, or [sub-]totalities of qualitatively different, or heterogeneous, ideas or «eide» — A.D.], cannot enter into any “community” with one another [i.e., are non-reductive, ‘‘‘nonlinear’’’, non-superpositioning”, non-additive, non-addable, or non-amalgamative / «asumbletoi»  A.D.].”

“Their monads are all of different kind [i.e., are categorially, ontologically, qualitatively unequalA.D.] and can be brought “together” only “partially”, namely only insofar as they happen to belong to one and the same assemblage, whereas insofar as they are “entirely bounded off” from one another...they are incapable of being thrown together, in-comparable [incapable of being counted as replications of the same quality of unit[y], of the same qualitative unit, or ‹‹monad››; incomparable quantitativelyA.D.] ... .”

“The monads which constitute an eidetic number, i.e., an assemblage of ideas, are nothing but a conjunction of eide which belong together.”

“They belong together because they belong to one and the same eidos [singular form of «eide»:  one particular ‘internal / interior seeing’, vision, or «ιδεα» — A.D.] of a higher order, namely a “class” or genos [akin to, and ancestor to, the grouping of multiple species under a single genus in classical biological ‘‘‘taxonomics’’’ or ‘‘‘systematics’’’ — A.D.].”

“But all will together be able to “partake” in this genos (as for instance, “human being”, “horse”, “dog”, etc., partake in “animal”) without “partitioning” it among the (finitely) many eide and without losing their indivisible unity only if the genos itself exhibits the mode of being of an arithmos [singular form of «arithmoi»:  a single assemblage, or multitude, of units / «monads» — A.D.].”

“Only the arithmos structure with its special koinon [“community” or “commonality” — A.D.] character is able to guarantee the essential traits of the community of eide demanded by dialectic; the indivisibility [a-tom-icity or un-cut-ability’ — A.D.] of the single monads which form the arithmos assemblage, the limitedness of this assemblage of monads as expressed in the joining of many monads into one assemblage, i.e., into one idea, and the untouchable integrity of this higher idea as well.  What the single eide have “in common” is theirs only in their community and is not something which is to be found “beside” and “outside”...them. ... .”

“The unity and determinacy of the arithmos assemblage is here rooted in the content of the idea..., that content which the logos [word; rational speech; ratioA.D.] reaches in its characteristic activity of uncovering foundations analytically.  

“A special kind of [all-of-one-kind, generic-units-basedA.D.] number of a particular nature is not needed in this realm, as it was among the dianoetic numbers [the «arithmoi monadikoi» — A.D.]..., to provide a foundation for this unity.  In fact, it is impossible that any kinds of numberarithmoi» — A.D.] corresponding to those of the dianoetic realm [the realm of ‘dia-noesis’ or of ‘«dianoia»’, i.e., of ‘pre-/sub-dialectical’ thinkingA.D.] should exist here, since each eidetic number is, by virtue of its eidetic charactereide»-character or «idea»-nature A.D.], unique in kind [i.e., qualitatively unique / distinct / heterogeneous in comparison to other «eide» — A.D.], just as each of its monads has not only unity but also uniqueness.  For each idea is characterized by being always the same and simply singular [\ additively idempotent, and \ also unquantifiable’, as per the axioms of the NQ «arithmêtikê» — A.D.] in contrast to the unlimitedly many homogeneous monads of the realm of mathematical number, which can be rearranged as often as desired into definite numbers. ... .”

“The “puremathematical monads are, to be sure, differentiated from the single objects of sense by being outside of change and time, but they are not different in this sense — that they occur in multitudes and are of the same kind (Aristotle, Metaphysics B 6, 1002 b 15 f.: [Mathematical objects] differ not at all in being many and of the same kind...), whereas each eidos is, by contrast, unreproducible [hence modelable by idempotent addition, or ‘non-addability’, and ‘non-quantifiability’A.D.] and truly one (Metaphysics A 6, 987 b 15 ff.:  Mathematical objects differ from objects of sense in being everlasting and unchanged, from the eide, on the other hand, in being many and alike, while an eidos is each by itself one only...).”

“In consequence, as Aristotle reports (e.g., Metaphysics A 6, 9876 b 14 ff. and N 3, 1090 b 35 f.), there are three kinds of arithmoi:  (1) the arithmos eidetikosidea-number, (2) the arithmos aisthetossensible number, (3) and “between”...these, the arithmos mathematikos or monadikosmathematical and monadic number, which shares with the first its “purity” and “changelessness” [here Aristotle reflects only the early, more ‘Parmenidean’, Plato, not the later, «Autokinesis» Plato — A.D.] and with the second [the third in hierarchical order] its manyness and reproducibility.  

“Here the aisthetic [sensible, i.e., ‘sense-able’, or sensuousA.D.] number represents nothing but the things themselves which happen to be present for aisthesis [for sense perceptionA.D.] in this number.”

“The mathematical numbers form an independent domain of objects of study [an independent «mathesis» in their own right A.D.] which the dianoia [the faculty of ‘pre-/sub-dialectical thinking’ A.D.] reaches by noting that its own activity finds its exemplary fulfillment in reckoning [i.e., account-giving] and counting ... .”

“The eidetic number, finally, indicates the mode of being of the noeton [that which exists for thought; that which thought “beholds”; the object of thought; the idea[l]-objectA.D.] as such  — it defines the eidos ontologically as a being which has multiple relations to other eide in accordance with their particular nature [i.e., in accord with their content A.D.] and which is nevertheless in itself altogether indivisible.”  

The Platonic theory of the arithmoi eidetikoi is known to us in these terms only from the Aristotelian polemic against it (cf., above all, Metaphysics M 6-9)...

[Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, Dover [NY:  1968], pp. 89-91, italic, bold, underline, and color emphasis added by A.D.]

William Riese describes the final known phase of Plato’s two systems of dialectical philosophy as follows, whose bifurcation is signaled in Platonic dialogue known as The Parmenides:

 The dialogues of the Socratic period provide that view of the world usually associated with Plato.”

“The period of transition and criticism, and the final synthesis, are little noted ...”

The Parmenides can be taken as signaling the change.  In this dialogue Socrates is unable to defend his Doctrine of Ideas [i.e., of the «Eide», herein denoted by E -- A.D.]. ...”

“Where the Republic and Phaedo stressed the unchanging nature of the soul, the emphasis in the Phaedrus is exactly reversed.  

 In this dialogue, the soul is the principle of self-motion [in Greek, «Auto-kinesis» -- A.D.], and we are told that the soul is always in motion, and what is always in motion is immortal.   

The difference now between spirit and matter is not changelessness in contrast with change, but self-motion, the essence of the soul, in contrast with derived motion.”

“The emphasis on self-motion is continued even in the Laws, Plato's final dialogue.

[William L. Riese, Dictionary of Religion and Philosophy: Eastern and Western Thought, Humanities Press, Inc. [New Jersey: 1980], pp. 442-443 [italic, bold, underline, and color emphasis added by A.D.]

By a dynamical mathematical model is usually meant an ideographical, ‘‘‘algebraic’’’, analytical analogue of a target reality, one that describes the quantitative variation, through time, of some metrics of the model constituents, within a fixed universe of such constituents, a fixed ontology, with such constituents constituting the presumptive “ontological commitments” of that model.   

Herein, by the term-phrase ‘[meta-dynamical] meta-model, we mean an ‘ontologically-dynamical, multi-ontological-epochs-spanning and -bridging, equational analogue of a target reality which, relative to a given epoch of the self-development of that reality, treated as being its present epoch, reconstructs the ontology of its past epochs, and also pre-constructs’, or “predicts”, the ontology of its future epochs.   

By a meta-equation we mean a super-equation, i.e., an equation ‘of second degree’, made up out of a heterogeneous multiplicity of equations ‘of first degree’; a super-equation «monad» or unit which is a meta-«monad»’, or meta-unit, made up out of a multiplicity, a sub-«arithmos», of equation-«monads», or of “mere” equation-units, as its internalized [sub-]«monads».

Thus, the generic dialectical meta-equation, or equation of the second qualo-fractal scale --

  |-|-|t    =    [ qa ]3t                                                                                                               

-- is made up out of  the following multiplicity/sequence of “mere” equations, equations of the first qualo-fractal scale, one qualitatively-distinct[ive] equation for each distinct value of t:

 |-|-|0   =    a;

 |-|-|1   =    a  +  b  +  c;

 |-|-|2   =    a  +  b  +  c  +  d  +  e  +  f  +  g  +  h  +  i, etc.

Our psychohistorical-dialectical meta-equation meta-model for the psychohistorical domain of Ancient Mediterranean Philosophy [M] can thus be expressed compactly, in a dialectical-ideographical form, ‘‘‘located’’’ 3 levels of meta-fractalscale down from the level 1 scale of h, the humanities «arithmos», in the E.D. Universal Taxonomy, as --
|-|-|2     =    [ H ]32      =     

H + P +  E  +  qEH  +   qEP  +  A   +   qAH +   qAP +   qAE
-- for which we have ‘‘‘solved’’’ explicitly the first 3 categories, and the 6th category, of the 9 categories total, and, in adialectical-pictographical form, for   

q6 <---]  A   =   qEE       qPH<PH>

connoting immanent, self-critique of  

E    =    qPH,

as --

V.  Suggestions for Reader Exploration.  The following further explorations of the psychohistorical-dialectical domain addressed in this vignette -- the domain of Ancient Occidental Philosophies -- via the tools of the NQ dialectical ideography, are recommended to our readers:

1.   Try your hand at “solving for” the five terms, generated by the meta-modelherein presented, that we left “unsolved” in our explication of that meta-model, viz.,  

qEH, [---> q4


qEP, [---> q5

as well as  

qAH, [---> q7


qAP, [---> q8


qAPH   =   qAE, [---> q9.  

 Hint:  ‘Categorograms’ of the form, e.g., ‘qYX’ are standardly interpreted, per the Encyclopedia Dialectica canon, as connoting the process/‘processor’ that produces the conversion of some X «monads» into Y «monads» -- or into «monads» which are hybrids of the X and the Y «monads»; of the assimilation of some of the Xs by Ys; of the appropriation of some Xs by Ys; of the adjustment of the existence of the Xs to the existence of the Ys, or of the critique of X by the Y, the critical review and evaluation of the merits and demerits of the X kind of ideas from the perspective of the Y kind of ideas, the correction of the X kind of ideas by means of the Y kind, and the theory of error of the X kind of ideas from the point-of-view of the Y kind.  

 In particular --

* qEH connotes the result of critique of Heraclitean philosophy in light of the “Socratic” philosophy of the «Eide»;

* qEP connotes the result of critique of Parmenidean philosophy in light of the “Socratic” philosophy of the «Eide»;

* qAH connotes the result of critique of Heraclitean philosophy in light of the Platonic philosophy of «Autokinesis»;

* qAP connotes the result of critique of Parmenidean philosophy in light of the Platonic philosophy of «Autokinesis»;

* qAE connotes the result of critique of “Socratic” «Eide» philosophy by the Platonic philosophy of «Autokinesis».

With the help of the hints above, can you identify specific, named or described doctrines of Plato, together with citations of passages in Plato’s Dialogues, which correspond with some or all of these five terms?   

For example, Plato criticizes the philosophy of Herakleitos, from the perspective of the “Socratic” philosophy of the «Eide», in the following terms: can that be a real thing which is never in the same state? ... for at the moment that the observer approaches, then they become other and of another nature, so that you cannot get any further in knowing their nature or state .... but if that which knows and that which is known exist ever ... then I do not think they can resemble a process or flux ...., in the dialogue Cratylus, in its paragraph 439, section e through paragraph 440, sections a-b.  [E. Hamilton, H. Cairns, editors, The Collected Dialogues of Plato, including the Letters, Princeton University Press [Princeton: 1989]. pp.  473-474].

2.   See if, by exploring other candidate «arché» for a Triadic Seldon Function meta-modelof this domain, you can construct a meta-modelwhich encompasses, e.g., the earlier philosophies of Thales [640-546 B.C.E.], and of Anaximander [610-547 B.C.E.], Anaximenes [588-524 B.C.E.], and/or Pythagoras [570-500 B.C.E.], as well as later philosophies, of Herakleitos [540-475 B.C.E.], Parmenides [515-450 B.C.E.], Democritus [460-370 B.C.E.], Plato [428-348 B.C.E.], and even of Aristotle [384-322 B.C.E.], in a single Triadic Seldon Function dialectical meta-model meta-equation.

3.   Determine whether or not you do better with a Dyadic Seldon Function meta-model, in covering a fuller range of the [psycho]history of Ancient Mediterranean Philosophy, than with the Triadic Seldon Function meta-modelthat you constructed for/in step 2 above.

Links to definitions of additional Encyclopedia Dialectica special terms deployed in the discourse above --

«arithmos aisthetos»

«arithmos eidetikos»

«arithmos monadikos»





dialectical categorial progression



The Human Phenome


immanent critique




ontological category






Seldon Functions

self-meta-monad-ization or self-meta-individual-izationor self-meta-holon-ization

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