Introductory Letter

This project appropriates the 'meta-fractal' self-similarity of those two different, successive scales of «problematique» so as to resume some final and neglected, zenith breakthroughs within the ancient Alexandrian flowering of humanity's ancient Mediterranean civilization, in relation to what has developed subsequently, and in a way which assimilates, also, the wealth of that subsequent development.

This has led to a rediscovery, in a higher, modernized, and less Parmenidean, more Heraclitean form, of Plato's lost «arithmos eidetikos», his "arithmetic of ideas" or "arithmetic of dialectics".

This «arithmos» is alluded to in his extant writings, but its full exposition is nowhere to be found in those portions of Plato's opus which survived the last Dark Age.

It has been 'psycho-archaeologically' reassembled in a seminal study, by Jacob Klein, as follows:

Plato may have already embarked upon an axiomatization of these three arithmetics, circa 380 B.C.E., even prior to Euclid of Alexandria's axiomatization of geometry, circa 300 B.C.E.:

We term 'Peanic' any progression of entities which conforms to the first four (Giuseppe) Peano postulates, circa 1889, which were formulated with the intent to axiomatize just the "standard natural numbers", but which are known to have "non-standard models".

'Dialectic' is a 'logic', or a ['Qualo-Peanic' 'Meta-Peanic'] 'pattern of what follows from what', more general than the "formal logic" of 'propositional followership'.

'Dialectic' generalizes about how natural populations, ensembles, systems, [sub-]totalities — both concrete, physical-'external' «arithmoi», and 'internal', human-conceptual «arithmoi», — change, including especially of how they change themselves.

'Dialectic' is about both '[allo-]flexion' or '[allo-]flexivity' — the 'bending', or '''alteration''', of the 'course of development' of one '[ev]entity' by the actions of others — and about 'self-re-flexivity', 'self-re-fluxivity', 'self-dialogue', 'self-controversion', self-activity, self-change, or "self-contra-kinesis" [in summary, about '''self-bending''': the 'self-induced', self-determining '''bending''' of the 'course of development' of an 'eventity' as a result of its own, immanent, '''inertial''', ''ballistic''', 'intra-dual', 'essence-ial', 'self-force'].

'Dialectic' is the name for the fundamental [and ever self-developing] modus operandi of nature, including that of human[ized] nature, but also including that of pre-human and extra-human nature.

'Dialectic' is about the subject/verb/object-identical meta-dynamic of 'quanto-qualitatively', 'quanto-ontologically' [self-]changing, [self-]developing, via-'metafinite'-singularity '[self-]bifurcating' 'meta-systems' or 'process-entities' ['eventities'], manifested in all levels, at all '[meta-]scales', for all '''orders''' of 'natural history', including that part of 'natural history' which we call Terran human history, and, by hypothesis, in the history of humanoid species generally, throughout this cosmos ['human-natur[e-]al history'].

For Plato, «Dialektike», 'dialectical thought-technology', as manifested in his «arithmos eidetikos», names a higher form of human cognition. It is higher than that of «Doxa», mere opinion. It is also higher than that of «Dianoia» or «Dianoesis»; higher than that which Hegel termed «Verstand», "The Understanding" [cf. Plato].

"Dialectical thought" names a higher stage of human cognitive development, a higher "state" of human [self-]awareness, a higher form of human self-identity, and of '''human subject-ivity''', beyond even those associated with the most advanced possible forms of axiomatic, deductive, mathematical logic, still «dianoetic» and partly sub-rational due to the frequent arbitrariness, authoritarianism, and dogmatism of their unjustified axioms and primitives:

"...disputation and debate may be taken as a paradigmatic model for the general process of reasoning in the pursuit of truth, thus making the transition from rational controversy to rational inquiry. There is nothing new about this approach. Already the Socrates of Plato's Theaetetus conceived of inquiring thought as a discussion or dialogue that one carries on with oneself. Charles Saunders Peirce stands prominent among those many subsequent philosophers who held that discursive thought is always dialogical. But Hegel, of course, was the prime exponent of the conception that all genuine knowledge must be developed dialectically. ... These conclusions point in particular towards that aspect of the dialectic which lay at the forefront of Plato's concern. He insisted upon two fundamental ideas: (1) that a starting point for rational argumentation cannot be merely assumed or postulated, but must itself be justified, and (2) that the modus operandi of such a justification can be dialectical. Plato accordingly mooted the prospect of rising above a reliance on unreasoned first principles. He introduced a special device he called "dialectic" to overcome this dependence upon unquestioned axioms. It is worthwhile to see how he puts [this] in his own terms:

Plato's writings do not detail in explicit terms the exact nature of this crucial enterprise of dialectic. Presumably we are to gain our insight into its nature not so much by way of explanation as by way of example  — the example of Plato's own practice in the dialogues." [Nicholas Rescher; Dialectics: A Controversy-Oriented Approach to the Theory of Knowledge; SUNY Press (Albany, NY: 1977); pages 46-48; bold, italic, underline, and color emphasis added by F.E.D.].

The procedure of formal proof, of deductively deriving theorems from axioms and postulates, is the exercise of «dianoesis» «par excellence». But the process of discovery, formulation, selection, refinement, and optimization of the individual axioms themselves, and of systems of axioms, resides beyond the «dianoetic» realm. Formal and mathematical logic provide it with no algorithmic guidance. That process belongs to the realm of dialectics.

“Zero was unknown as a number, and one also was not counted as a number, the first number being the duas — two.” [J. O. Urmson, The Greek Philosophical Vocabulary, Duckworth & Co., Ltd. [London: 1990], pp.31-32].

“From the Pythagoreans … — who consider number to be the first principle — number played a great role in metaphysics, especially in Plato’s unwritten doctrines, involving obscure distinctions of e.g. ‹‹sumblêtoi›› and ‹‹asumblêtoi›› — addible and non-addible — numbers.” [Urmson, ibid., emphasis added by F.E.D.].

“arithmós:  number (see also arithmos eidetikos and arithmos mathematikos)

…    3.  The most perplexing aspect of ancient number theory is Aristotle’s repeated assertions that Plato taught that the eide were numbers (e.g. Meta. 987b), a position that must be distinguished from 1) the existence of the eide of numbers (see arithmos eidetikos) and 2) the existence of the “mathematicals” as an intermediate grade of being (see mathematika, metaxu). But nowhere in the dialogues does Plato seem to identify the eide with number. To meet this difficulty some have postulated a theory of later “esoteric” Platonism known to Aristotle (but see agrapha dogmata), while others have attempted to see the emergence of the eide-arithmos theory described in such passages as Phil. 25a-e, the reduction of physical objects back to geometrical shapes in Tim. 53c-56c (see stoicheion), and the increasing stress on a hierarchy among the Forms (see Soph. 254d and genos, hyperousia), which, according to Theophrastus, Meta. 6b, would suggest the descending series archai (i.e., monas/dyas or peras/apeiron, qq.v.), arithmoi, eide, aistheta. Still others say that Aristotle either deliberately or unknowingly confused the position of Plato with those of Speusippus and Xenocrates (see mathematika).” [F. E. Peters, Greek Philosophical terms:  A Historical Lexicon, NYU Press [NY: 1990], pp. 25-26].

“On the testimony of Aristotle dialectic was an invention of Zeno the Eliatic, probably to serve as a support for the hypothetical antinomies of Parmenides ... But what was a species of verbal polemic (what Plato would call “eristic” or disputation...) for the Eliatics was transformed by Plato into a high philosophic method. The connecting link was undoubtedly the Socratic technique of question and answer in his search for ethical Definitions…, a technique that Plato explicitly describes as dialectical (Crat. 390c). …With the hypostatization of the Socratic definitions as the Platonic eide … the role of dialectic becomes central and is the crown of the ideal curriculum described in the Republic:  after ten years devoted to mathematics the philosopher-to-be will devote the years between thirty and thirty-five to the study of dialectic. …

What is dialectic?  The question is not an easy one since Plato, as usual, thought about it in a variety of ways. There is the view of the Phaedo and the Republic, which envisions dialectic as a progressively more synoptic ascent, via a series of “positions” (hypotheseis, q.v.; the Theory of Forms is one such in the Phaedo 100b), until an ultimate is reached (Phaedo 101d, Rep. 511e).  In the Republic, where the context of the discussion is confessedly moral, this “unhypothesized principle” is identified with the good-in-itself (auto to agathon; Rep. 532a-b) that subsumes within itself all of the lower hypotheses (ibid., 533c-d) [cf. the Hegelian core concept of dialectic, named by the German word ‹‹aufheben›› — F.E.D.] … If the dialectic of the Phaedo and the Republic may be described as “synoptic” …, that which emerges from the Phaedrus onward is decidedly “diacritic”… it is introduced in Phaedrus 265c-266b (compare Soph. 253d-e) and consists of two different procedures, “collection” (synagogue, q.v.) and “division” (diairesis, q.v.), the latter process being amply illustrated in subsequent dialogues like the Sophist, Politicus, and Philebus. The earlier dialectic appeared similar to the operations of eros (q.v.) [recall Herbert Marcuse’s comment, in his Reason and Revolution, to the effect that '''eros is the force that binds matter together into ever higher unities'''-- F.E.D.], but here we are transported into an almost Aristotelian world of classification through division; ascent has been replaced by descent. While it is manifest that we are here still dealing with ontological realities, it is likewise clear that a crucial step has been taken along the road to a conceptual logic. The term [i.e., the terminus – F.E.D.] of the diairesis is that eidos which stands immediately above the sensible particulars (Soph. 229d), and, while this is “really real” (ontos on) in the Platonic scheme of things, it is significant that the same process ends, in Aristotle, in the atomon eidos, the infima species in a logical descent (De an. II, 414b)…” [Peters, ibid., pp. 36-37].

“diairesis: separation, division, distinction

1. Division, a procedure that did not interest Socrates since the thrust of his enquiry was toward a single eidos (see epagoge), becomes an important feature in the later dialogues where Plato turns his attention to the question of the relationship between eide.  Expressed in terms of Aristotelian logic diairesis is part of the progress from genus to species; but, as is clear from a key passage in the Parmenides, where he first puts the question (129d-e), Plato did not see it as a conceptual exercise. The dialectical search of which diairesis is part has as its object the explication of the ontological realities that are grasped by our reflection (logismos).

2. The pursuit of the interrelated eide begins with an attempt at comprehending a generic form (Phaedrus 265d); this is “collection” (synagoge, q.v.).  It is followed by diairesis, a separation off of the various eide found in the generic eidos, down to the infima species (Soph. 253d-e). Plato is sparing of details in both the theory and the practice of synagoge, and, while the Sophist and the Politicus are filled with examples of diairesis, there is relatively little instruction on its methodology. We are told, however, that the division is to take place “according to the natural joints” (Phaedrus 265e). What these are becomes clearer from the Politicus:  they are the differences (diaphorai, q.v.) that separate one species from another in the generic form (Pol. 262a-263b, 285b).

3.  The method of division raises certain serious questions, so serious, indeed, that they might very well shake confidence in the existence of the eide. …  Do the species constitute the genus or are they derived from it? …” [Peters, ibid., pp. 34-35].

“diairien (in past tense, dielein), diairesis:  to divide, division, used in many contexts in Greek as in English. In philosophy particularly the logical division of a genus into species. In the Phaedrus and the Sophist Plato speaks of a method of sunagôgê — collection – and diairesis — division — as the supreme method of philosophy: … and, Phaedrus, I myself am a lover of divisions and collections in order to become able to speak and think (Pl. Phaedrus 266b); … — unless one is capable of dividing things and subsuming each thing individually under a single form, one will never become skilled in discussion to the limit of human capacity (Pl. Phaedrus 273d): … — a longstanding laziness about dividing genera into species (Pl. Soph. 267d).” [Urmson, ibid., pp. 39-40].

“sunagein: to collect; sunagôgê:  the action of collecting. Non-technically: … we shall bring together the brides and the bridegrooms (Pl. Rep. 459e).  Also used as a technical term by Plato, particularly in the Sophist and the Phaedrus, where the contrary of sunagôgê is diairesis, division: … — I am myself, Phaedrus, a lover of these divisions and collections (Pl. Phaedrus 266b). Collection appears to be bringing together under a single genus a variety of things which are then to be divided formally into species and sub-species: … — to survey under one form things that are scattered in many areas (Pl. Phaedrus 265d).” [Urmson, ibid., pp. 158-159].

“synagôgé: collection

The Platonic type of “induction” (for the more normal type of induction, i.e., a collection of individual instances leading to a universal, see epagoge) that must precede a division (diairesis) and that is a survey of specific forms (eide) that might constitute a genus (Phaedrus 265d, Soph. 253d).  An example is Soph. 226a, and the process is also suggested in Rep. 533c-d, and Laws 626d…” [Peters, ibid., p. 188].

“eídos:  appearance, constitutive nature, form, type, species, idea

…  12. At various points in the dialogues Plato seems to grant preeminence to one or other [sic] of the eide.  Thus, both the Good (Rep. 504e-509c) and the Beautiful (Symp. 210a-212b) are thrown into relief, to say nothing of the notorious hypotheses of the One in the Parmenides (137c-142; see hen, hyperousia). But the problem of the interrelationship, or, as Plato calls it, “combination” or “communion” (koinonia), and, by implication, of the subordination of the eide is not taken up formally until the Sophist.  It is agreed, again on the basis of predication, that some eide will blend with others and some will not, and that it is the task of dialectic to discern the various groupings, particularly through the diacritic method known as diairesis (q.v.; Soph. 253b-e). …  [Peters, ibid., p. 49, emphasis added by F.E.D.].

…  8. Though the eide are the centerpiece of Platonic metaphysics, nowhere does Plato undertake a proof for their existence; they first appear as a hypothesis (see Phaedo 100b-101d) and remain so, even though subjected to a scathing criticism (Parm. 130a-134e).  They are known, in a variety of methods, by the faculty of reason (nous; Rep.532a-b, Tim. 51d). One such early method is that of recollection (anamnesis, q.v.), where the individual soul recalls the eide with which it was in contact before birth (Meno 80d-85b, Phaedo 72c-77d; see palingenesia). Without the attendant religious connotations is the purely philosophical method of dialektike (q.v.; see Rep. 531d-535a; for its difference from mathematical reasoning, ibid., 510b-511a; from eristic, Phil. 15d-16a).  As it is first described the method has to do with the progress from a hypothesis back to an unhypothesized arché (Phaedo 100a, 101d; Rep. 511b), but in the later dialogues dialektiké appears as a fully articulated methodology comprising “collection” (synagoge, q.v.) followed by a “division” (diairesis, q.v.) that moves, via the diaphorai, from a more comprehensive Form down to the atomon eidos.  Finally, one may approach the eide through eros (q.v.), the desiderative parallel to the earlier form of dialectic (see epistrophe).” [Peters, ibid., pp. 47-48].

"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. ... 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, 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); pages 442-443, emphasis added by F.E.D.].

[Phaedrus]:  “But that which while imparting motion is itself moved by something else can cease to be in motion, and therefore can cease to live; it is only that which moves itself that never intermits its motion, inasmuch as it cannot abandon its own nature; moreover this self-mover is the source and first principle of motion for all other things that are moved.  Now a first principle cannot come into being, for while anything that comes to be must come to be from a first principle, the latter itself cannot come to be from anything whatsoever; if it did, it would cease any longer to be a first principle. Furthermore, since it did not come into being, it must be imperishable, for assuredly if a first principle were to be destroyed, nothing could come to be out of it, nor could anything bring the principle itself back into existence, seeing that a first principle is needed for anything to come into being.

The self-mover, then, is the first principle of motion, and it is as impossible that it should be destroyed as that it should come into being; were it otherwise, the whole universe, the whole of that which comes to be, would collapse into immobility, and never find another source of motion to bring it back into being.” [Plato, The Collected Dialogues, E. Hamilton, H. Cairns, editors, Princeton U. Press [Princeton:  1989], Phaedrus, 245c-e, pp. 492-493, italic and bold-italic colored text emphasis added by F.E.D.].

[Laws]:  “When we have one thing making a change in a second, the second, in turn, in a third, and so on – will there ever, in such a series, be a first source of change? Why, how can what is set moving by something other than itself ever be the first of the causes of alteration? The thing is an impossibility. But when something which has set itself moving alters a second thing, this second thing still a third, and the motion is thus passed on in course to thousands and tens of thousands of things, will there be any starting point for the whole movement of all, other than the change in the movement which initiated itself? …

Suppose all things were to come together and stand still – as most of the party have the hardihood to affirm. Which of the movements we have specified must be the first to arise in things?  Why, of course, that which can move itself; there can be no possible previous origination of change by anything else, since, by hypothesis, change was not previously existent in the system. Consequently, as the source of all motions whatsoever, the first to occur among bodies at rest and the first in rank in moving bodies, the motion which initiates itself we shall pronounce to be necessarily the earliest and mightiest of all changes, while that which is altered by something else and sets something else moving is secondary.” [ibid., Laws, 10.894e-10.895b, pp. 1450, bold-italic, underlined, and colored text emphasis added by F.E.D.].

1.    of ‘‘‘ideas as ‘unaddable’ numbers’’’, & of ‹‹dialektiké›› as an ‘‘‘arithmetic of ideas’’’; the arithmetic of the ‹‹arithmoi eidetikoi››;

2.   of ‹‹dialektiké›› as the highest philosophic method, one similar in its operation to that of eros; a synoptic method, a method of ascent,  via a series of “positions”, or hypotheses, until an ultimate is reached, that subsumes within itself all of the lower hypotheses;

3.   of ‹‹dialektiké›› as a diacritic method, a method of descent — of synchronic ‘ideo-systematics’, ‘ideo-taxonomics’, or ‘ideo-meta-genealogy’, for the correct determination of the … «gene», the «species», and the sub-«species» …, etc., of the «eide»… — a method composed of two distinct, opposite procedures, or movements; first by one of “collection”[«synagoge»], into «gene», followed, second, by one of “division” [«diairesis»], into “classes” — into «species», sub-«species» …, etc. — of the fundamental «ιδεα^ς» that, per Plato, undergird this «kosmos», and;

4.   of ‹‹arché kinesis›› as ‹‹auto kinesis››.

(i) the non-self-evidence of appropriate and optimal axioms generally; the exercise of choice and skillful design required in their development and selection, and the abounding 'alternativity' which that activity confronts;

(ii) the axioms-dependence or assumptions-relativity of all formal proofs, hence of all formal "truths";

(iii) the logical 'equi-coherence' of non-standard models of "first order" axioms-subsystems with respect to their standard models;

(iv) the independence or Gödel-undecidability of key axioms of "higher-than-first-order" axioms-systems with respect to the rest of the axioms, hence the logical 'equi-coherence' of alternative axioms-systems, built on contraries of those key axioms, and especially;

(v) 'The Gödelian Dialectic'; the psycho-historical, «aufheben»/evolute-cumulative progression of de facto axioms-system within the social and socio-cognitive, '''Phenomic''' progression of a human species.

"It can be shown that any formal system whatsoever — whether it is based on the theory of types or not, if only it is free from contradiction — must necessarily be deficient in its methods of proof. Or to be more exact: For any formal system you can construct a proposition — in fact a proposition of the arithmetic of integers — which is certainly true if the system is free from contradiction but cannot be proved in the given system [the foregoing summarizes Gödel's "First Incompleteness Theorem" — F.E.D.]. Now if the system under consideration (call it S) is based on the theory of types, it turns out that exactly the next higher type not contained in S is necessary to prove this arithmetic proposition, i.e., this proposition becomes a provable theorem if you add to the system the next higher type and the axioms concerning it." [Kurt Gödel; "The Present Situation of the Foundations of Mathematics (*1933o)", in S. Feferman, et. al., editors.; Kurt Gödel: Collected Works (Volume III: Unpublished Essays and Lectures); Oxford University Press (NY: 1995); page 46; bold, italics, underline, and color emphasis added by F.E.D.].

"If we imagine that the system Z  [a formal, logical, propositional-/predicate-calculus system inclusive of "Natural" Numbers'  Arithmetic, not the full system of the positive and negative Integers, and zero [which is both, [or neither] positive [n]or negative], standardly also denoted by Z — F.E.D.] is successively enlarged by the introduction of variables for classes of numbers, classes of classes of numbers, and so forth, together with the corresponding comprehension axioms, we obtain a sequence (continuable into the transfinite) of formal systems that satisfy the assumptions mentioned above, and it turns out that the consistency (ω-consistency) of any of these systems is provable in all subsequent systems. Also, the undecidable propositions constructed for the proof of Theorem 1 [Gödel's "First Incompleteness Theorem" — F.E.D.] become decidable by the adjunction of higher types and the corresponding axioms; however, in the higher systems we can construct other undecidable propositions by the same procedure. ...To be sure, all the propositions thus constructed are expressible in Z (hence are number-theoretic propositions); they are, however, not decidable in Z, but only in higher systems..." [Kurt Gödel; On Completeness and Consistency (1931a), J. van Heijenoort, editor; Frege and Gödel: Two Fundamental Texts in Mathematical Logic; Harvard University Press (Cambridge: 1970); page 108; bold, italic, underline, and color emphasis and [square-brackets-enclosed commentary] added by F.E.D.].

"... The Gödel sentence φ... asserts its own undeducibility from the postulates....Deformalizing φ... we see that under the standard interpretation it expresses a fact of the form [for every n-ary list of number-components of x such that each number-component is a member of the set of 'diophantine' or "Natural" Numbers in use — F.E.D.] ...ƒx ≠­ gx... , where ƒ and g are n-ary polynomials....An equation ƒx = gx, where ƒ and g are two such polynomials, is called diophantine [see below for further information regarding Diophantus of Alexandria  — F.E.D.] .... By a solution of the equation we mean an n-tuple α of natural numbers such that ƒα = gα... . So φ... asserts the unsolvability of the...equation ƒx = gx, and the proof of [Gödel's "First Incompleteness Theorem" — F.E.D.] produces... a particular diophantine equation that is really unsolvable, but whose unsolvability cannot be deduced from the postulates..." [Moshé Machover; Set Theory, Logic, and their Limitations; Cambridge University Press (Cambridge: 1996); pages 268-269].