Full Title -- Dialectic: The Journey of the Progression of the <<Aporia>>
The following selection from the Background section of The Goedelian Dialectic of the Standard Arithmetics, Part I., explores the meaning of the term "dialectic" in the sphere of the human phenome, in relation to the <<aporia>> -- the impasses or quandaries
-- that arise in the struggle to coherently define the key concepts of
even supposedly "pure" idea-systems such as those of philosophical
concepts, or of the standard axiomatic systems of arithmetic.
The full essay can be accessed via the following URLs --
"Dialectic:The Journey of the Progression of the «Aporia».
We will endeavor to instill, herein, for readers unversed in dialectic,
the minimum background in that cognitive stage requisite for an apt
appropriation of our narration of ‘The Dyadic Seldon Function Equational
Meta-Model of our Meta-Systematic Dialectical Presentation of the
Progression of the Standard Arithmetics’ in the core section of this
essay -- the same ‘Meta-Model’ introduced in the background sub-section
immediately preceding this one.
We will here present, with our commentary, an extended excerpt from an
excellent source, just-published, on ‘The [Psycho]Historical Dialectic
of Modern Science’: Newton, Maxwell, Marx, by Thomas K. Simpson, Green Lion Press [Santa Fe: 2012].
Despite the ‘extendedness’ of this excerpt, it provides one of the most succinct introductions to ~2500
years of Terran humanity’s [psycho]history of conscious dialectic that
we have so far found, located toward the end of the final section of the
above-cited book, the section on the work of Karl Marx.
Here it is:
“We have spoken earlier of two models of “science” -- that which
aims to save the appearances, and was exemplified by Ptolemy, and
another, exemplified by Newton and The Principia, which seemed the model that Marx follows in Capital in its aspect as formal science.
Now, however, I am proposing a third, for I believe that the dialectical reading of Capital is not less “scientific” than the formal one, but rather that it very much enlarges the sense and scope of “science”.
We would be right, I believe, in speaking of Capital in its larger aspect as a work of dialectical science.
Though this thought cannot be adequately developed here, I would like to suggest what it might mean.
To do so, however, we must take a moment to reflect on the notion of “dialectic” itself.
For though I have often referred to the concept in the course of this
study, and made certain claims about it in passing, I have not explained
what I understand it to mean.” [op. cit., pp. 261-262].
Thus, Dr. Simpson introduces his account of the history and meaning of
the word “dialectic”, in the context of a concept of “dialectical
As regular readers of the www.dialectics.org web site will be aware, that phrase names the stage of science to which we refer our work as well.
Dr. Simpson next notes the [long] history of the concept of “dialectic”
in the tradition of the Occidental hemisphere of the Terran Human[oid]
Phenome, and the way in which that history exhibits a [psycho]historical
dialectic of its own -- “the dialectic of the dialectic[ itself]” --
another theme which visitors of this website will recognize as seminal
for F.E.D. [ dialecticádialecticñ link ]:
“Dialectic has its own history in our Western experience; we
might speak of its own “dialectical” development -- or of “the dialectic
of the dialectic.”
It will suffice for the moment if we take just three great benchmarks of
dialectic as exemplars of this unfolding concept: the Platonic
dialogues, of course; the Hegelian dialectic; and the dialectic of Capital.” [op. cit., p. 262].
[Link to Publisher’s Page for Newton, Maxwell, Marx].
Aside from the writings of Heraclitus -- only shreds of which survived
the turmoils of the last Dark Ages -- Dr. Simpson’s selection of
dialectic’s “benchmarks” is one with which F.E.D. heartily agrees.
However, the reader should note that it is not clear, in Dr. Simpson’s account, that he would agree entirely with F.E.D.’s delineation of four ‘historical «species»’ of dialectic, as constituting “the dialectic of the dialectic[ itself]”.
Dr. Simpson’s account continues as follows: “... the term
“dialectic” suggests ... a certain structure of inquiry, and we need to
consider the relation between such a real question on the one hand, and
the pattern of dialectic on the other: not only the vividness of the
human questions that they ask, but a certain common form links Plato,
Hegel, and Marx.
As we say this, however, it is very important to avoid possible
misunderstanding: though the thread that links these stages of
dialectic is very real, and fundamental in particular to our
understanding of Marx, to point to this common principle is not at all
to assert that dialectic is not very much transformed in its passage
from one stage to the next.” [op. cit., p. 262].
In the Encyclopedia Dialectica shorthand, this pattern of dialectic, common to Plato, Hegel, and Marx, can be rendered generically as:
x ----) x(x) = ~(x) = (x^2) = (x + y) ----) (x + y)^2 = (x + y)( (x + y) )
= ~( (x + y) ) = ( (x + y)^2 ) = (((x + y) + z) + ... )
-- or, in words, as: ‘ “thesis” [(x)] becomes thesis self-reflected & immanently [self-]critiqued = ‘antithesis-sum’ [(x + y)], which in turn becomes ‘antithesis-sum’ itself self-reflected and immanently [self-]critiqued = ‘synthesis-sum +...’ [(((x +y) + z) + ...)] '.
But, before formulating any such pattern of dialectic in general, Dr. Simpson cites and summarizes Plato’s dialogue The Meno, as a paradigmatic example thereof:
“The dialogue Meno serves well as a paradigm of the dialectical motion in its Socratic mode ... To focus, then, for a moment on the Meno
as a paradigm of the dialectical form: the opening question is
developed in a variety of artful ways that steadily reduce this brash
young general, initially confident enough in the world’s ways, to a
state of what may be serious wonder and concern.
He meets unexpected difficulty in defining virtue (“excellence”), though
he had evidently never before doubted that he was himself a living
model of it. His failure leads him to a certain, perhaps petulant,
despair, and it is significant that here the dialogue comes to a dead
stop -- at its effective center -- with an outcry from Meno that seems
in its own terms unanswerable ...
This point of death of the argument -- which appears, I am tempted to
say at the virtual center of all dialectic -- is often referred to in
Greek as the aporia -- the sticking point, the passage of no passage, the point of no return for the argument.” [op. cit., pp. 262-264].
What is at stake in the ‘self-dialogue meta-model’ to be narrated and
explicated in the core section of this essay, is the definition of
[counting] number, and its stagéd incompletenesses and self-antitheses, starting from the definition of “cardinal number” that is embodied in the N axioms-system for the “Natural” Numbers as “first thesis”.
The succession of «aporia», or of quandaries, that are both evoked and transcended per this ‘meta-model’ may seem muted in their conceptual and emotional intensity, relative to those encountered, e.g., in Plato’s Meno, with regard to what constitutes “the good life” for humans individually, and, e.g., in Marx’s Capital, regarding the good and the bad of the life of the human -- historical -- «species» collectively.
It will remain so muted, at least until the connexion
between the history of the concept of number -- as a psychohistorical
barometer of the initially unconscious historical emergence of the
‘‘‘Law of [Capital-]Value’’’ in human praxis, and, consequently, of that ‘‘‘Law’s’’’ shaping of the quality, for good and for ill, of the life of the human «species»,
as an unconsciously-legislated, and continually ‘re-legislated’,
self-imposition, by that life, upon that life itself, both collective
and individual -- becomes a connexion that is far more widely grasped within the human «species» than it is at present.
This effective affective muting will be amplified by the fact that the readers of this essay, in their majority, will already know the ways around and beyond these «aporia»
of the concept of [counting] number that their ancestors have long
since already discovered -- sometimes only after long and arduous
Such readers will thus tend not to feel the bite of each quandary -- that their ancestors did once feel, when such a quandary was still unresolved -- here, in this present-mathematical-viewpoint-based, essentially synchronic, ‘ideo-systematic’, ‘ideo-taxonomic’ presentation.
Only the narration of a diachronic, [psycho]historical-dialectical ‘meta-model’, of the actual historical order and content of the successive human-phenomic memes of number,
with careful exegesis of well-chosen selections from their extant
‘psycho-artefacts’ -- their written controversies, etc. -- can serve to
revivify and to re-illuminate the heat and the rancor of past conceptual
struggles -- can overcome the illusory muting affect induced by our
already knowing the historic resolutions of past «aporia».
Such a historical narration is not the purview of the present essay, but is, in part, that of Dialectical Ideography.
Having so descried, and so described, the «aporia» that irrupts in and as the middle of the Meno, Dr. Simpson leaves behind the Meno’s specificity, and formulates the common pattern of dialectic in its generality, thusly:
“We cannot here trace that way by which Socrates at once opens a path for Meno -- and, we must add, for us. ...
The aporia of the dialectic will always lead us into the darkest
obscurity; but if all goes well, we will emerge empowered with knowledge
we had not known we possessed.
To be schematic about this, and as an aid in tracing something of this
same pattern in Hegel and Marx, we may say that there are three parts to
every dialectical motion.
I. The opening question, a real question, which takes form
through an intense searching in the mode of questions and answers, not
yet fully articulate;
II. The clarifying argument to the point of contradiction and
despair; the question becomes articulate but, at the same time, leads to
III. The passage beyond the aporia, through yet more
serious questioning, which yields whatever knowledge is humanly
possible; not, however, in the mode of syllogistic consequence, but
drawing on some larger source of human intuition, in the form of image,
myth and mystery.” [op. cit., pp. 264-265].
We would characterize the «genos» of the items in Dr. Simpson’s list of larger sources of human intuition under the category which one of F.E.D.’s early mentors, Dr. Charles Musés, used to call «Symbólos», «Symballein», or «Symbolon» -- that of The Symbolic.
This is a category which also includes the mathematical in its largest sense.
In the core narrative of this essay, it is new mathematical symbolizations, symbolizations of new number ‘ideo-ontology’; of new kinds of number -- and, therefore, of new ‘ideo-technology’, making possible useful calculations which could not previously, on the “pre” side of each such ideo-systematic «aporia», have been carried out -- that objectify our ancestors’ subjective escape from the dialectical «aporia» of «arithmêtikê»; from the dialectical «aporia» of the concept of “cardinal number”.
Definitions for the terms ‘‘‘ontology[-in-general]’’’, ‘physio-ontology’, and ‘ideo-ontology’, in their Encyclopedia Dialectica senses, can be reviewed via the following link --
Link to E.D. Definition of the Term '''Ontology'''.
¿But, given the above rendition of the pattern of
dialectic in general, what do we mean by ‘The Gödelian Dialectic’
specifically -- the term at the root of this essay’s title, ‘The Gödelian Dialectic of the Standard Arithmetics’ -- and how does this specific, Gödelian, dialectic, fit the general pattern outlined by Dr. Simpson?"