F.E.D. Dialectics: 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 www.dialectics.info website.

The URLs for that version are the following --

http://www.dialectics.info/dialectics/Welcome.html

http://www.dialectics.info/dialectics/Vignettes.html http://www.dialectics.org/dialectics/Vignettes.html

http://www.dialectics.info/dialectics/Welcome_files/Aoristos_Dyosphainthos,v.2.0,F.E.D._Vignette_12,The_Dialectic_of_Ancient_Mediterranean_Philosophy_Through_Plato%27s_Philosophical_Developments,04MAY2013.pdf

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

Enjoy!

Regards,

Miguel

F.E.D. Vignette #12 --

The [Psycho]Historical Dialectic of

Ancient Mediterranean Philosophy

through Plato’s Philosophical Developments

by Aoristos Dyosphainthos

Author’s 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 all of the way through to 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-dialectical ‘ideo-meta-dynamics’ of 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 --

http://www.dialectics.info/dialectics/Glossary.html

https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/ClarificationsArchive.htm

-- 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 --

«arché»

https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Arche/Arche.htm

«arithmos» and «arithmoi»

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Arithmos/Arithmos.htm

«aufheben»

https://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Aufheben/Aufheben.htm

Historical Dialectics

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/HistoricalDialectics/HistoricalDialectics.htm

_NQ dialectical arithmetic/algebra

http://www.dialectics.info/dialectics/Correspondence_files/Letter17-06JUN2009.pdf

Psychohistorical Dialectics

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/PsychohistoricalDialectics/PsychohistoricalDialectics.htm

-- 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-generated’ progression 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-numbers’ of 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-subsumption’ of 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 q_N.

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’’’ specific “Natural Numbers”, as specific ‘ordinal quantifiers’, to the generic ordinal qualifier symbol ‘q’, by ‘subscripting’ those specific “Natural 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 entailment’ can 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-reflexion’ via 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 ]v^h, 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 >3^t,

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 by ‘qa[---> 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’ [ http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf ], a ‘self-iterated’, cumulative recurrence of dialectical “negations 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-category’ or ‘‘‘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 > > >’,

with

< 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-equation’ constructed 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-model’ in 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

qHH,

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’’’,

qHH,

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’’’,

qPH,

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”,

qPH,

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 numbers [«arithmoi»; assemblages of units — A.D.] with which the arithmetician deals, the arithmoi mathematikoi or monadikoi [abstract, generic, idealized, qualitatively-identical, homogeneous “monads” or [ideal[ized], abstract qualitative units — A.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-toms — A.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 unequal — A.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 quantitatively — A.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; ratio — A.D.] reaches in its characteristic activity of uncovering foundations “analytically”.

“A special kind of [all-of-one-kind, generic-units-based — A.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 number [«arithmoi» — A.D.] corresponding to those of the dianoetic realm [the realm of ‘dia-noesis’ or of ‘«dianoia»’, i.e., of ‘pre-/sub-dialectical’ thinking — A.D.] should exist here, since each eidetic number is, by virtue of its eidetic character [«eide»-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 “pure” mathematical 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 eidetikos — idea-number, (2) the arithmos aisthetos — sensible number, (3) and “between”...these, the arithmos mathematikos or monadikos — mathematical 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 sensuous — A.D.] number represents nothing but the things themselves which happen to be present for aisthesis [for sense perception — A.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]-object — A.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 ]3^t

-- 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-fractal’ scale 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 a ‘dialectical-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-model’ herein presented, that we left “unsolved” in our explication of that ‘meta-model’, viz.,

qEH, [---> q₄,

and

qEP, [---> q₅,

as well as

qAH, [---> q₇,

and

qAP, [---> q₈,

and

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: “...how 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-model’ of this domain, you can construct a ‘meta-model’ which 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-model’ that you constructed for/in step 2 above.