The experience of this material leaves with me most strongly the impression that a new conceptual universe is opening before the human race; that the work here recounted is being evoked by deep and urgent inner promptings, arising by pathways mostly invisible to the prevailing consciousness, out of a context of the unprecedented global inflorescence of human socialized praxis, and its rapidly generalizing emergency within the capitalist integument; that this work is an expression of the last upsurgence of the productive forces within capital, as well as of the increasing strangulation and decadence of the productive forces within capital, and that these conceptual developments bear within the seeds, rapidly germinating, of socialist revolution at the level of the "superstructure". It is also this sense, this spirit of the times, which I hope to convey here, together with something of its emerging conceptual substance. For, whether we know it or not, we are now living in what will become -- if Humanity is to be born alive from the womb of its Prehistory -- the cradle of the Socialist Renaissance, that is, of the most fertile era of human creativity in all of recorded time.
In the section immediately following, we seek to situate the concept of dialectics in Marxian theory as a whole, to provide a general orientation to the relevance of all that follows for Socialist theory and for the self-comprehension of the revolutionary process.
1. Dialectics and Marxian Theory
"The outstanding achievement of Hegel's Phenomenology
-- the dialectic of negativity as the moving and creating principle --
is, first, that Hegel grasps the self-creation of man as a process...
and that he, therefore, grasps the nature of labour, and conceives objective man (true, because real man) as the result of his own labour." c1 -- Karl Marx, Economic-Philosophic Manuscripts
(a) Marx's Reflexive Functions
No aspect of Marxian
theory is less understood, or more fundamental, than Dialectics. That
dialectics is, in fact, the throbbing heart of Marxian theory
is widely acknowledged, both by those who consider themselves Marxians,
and those who do not. But this assertion has become, in most quarters,
more a mechanical recitation than a vital declaration. What, in fact,
does it mean?
We can get at this question by locating for ourselves what might be called the 'fundamental theorem' of Marxian theory; the unitary conception which implicitly contains the entirety of the theory, and then by seeing what, if anything, it has to do with Dialectics. This founding premise may be stated as follows: the central consequence of human activity in the world is (or has been up to now) the continued existence of humanity itself -- re-existence. reproduction, or self-reproduction. That is, the consequence of human activity is human activity itself again, and more of itself (including more developed qualities of itself).
Every human being alive today has the evidence for this proposition ready to hand in the most immediate possible form; that provided by his or her self itself; by the self-presentation of his or her own consciousness -- the self-evidence of human existence. The recognition of this evidence as such, of course, presupposes the related realization that a human identity, a self, is not an isolatedly self-subsistent being, an independent "atom" of self-consciousness, unrelated to the totality of humanity, but rather a society-subsistent one, supported by the social whole -- and by, mediated through that whole, the process of the entire cosmos, or "Nature", the full totality of material being c2 -- having no existence apart from that whole. 'I' am capable of continuing only through a constant, kaleidoscopic 'energy-exchange' relationship with this totality. The very possibility of my present quality of 'self' is a social product, a product and 're-product' of the present quality of society. Nor does the structure of the cosmic process guarantee
continued existence of itself. Inexistence is also quite possible.
Apart from the constant, conscious -- and effective, appropriate -- effortful intervention of human beings in this cosmic process, as part of it, all these 'I's would cease to exist.
Now, we can certainly exhibit this 'fundamental theorem' of social reproduction in more compact form:
Humanity makes itself.
Humanity produces humanity.
(reflexive function set 1)
Note how other key
concepts of Marx's model of social evolution can be oriented and
resolved in terms of this concept, of the self-production of man. The
'productive forces' concept refers to the force or 'forcefulness' of
this self-production, the strength of the force of social reproduction, or the self-productivity of humanity. The 'production-relations' concept refers to the internal social self-connexions carrying this self-production process and determining its rate; to the internal ratios or 'relatios' (verhaltnisse; rapport) expressing its organization for reproduction. The concept of social production refers primarily to this 'production of society (by society)',
and only secondarily, by way of the exploited ambiguity of the phrase,
to other conceptual moments usually more readily associated with its
meaning (such as: associated production, or production in society, in association; production by society, of particular objects, etc.) c3. 'History' is nothing but the historical labor of the self-construction of man. Not merely discrete artifacts, objects of utility, but humanity as a totality, is the product of the labor of history, as well as its agent. The concept of self-production also intersects those of social-evolutionary epoch and mode of production. The self-production of man is the continuum, the overall pattern and unbroken thread which connects each of the successive social formations and social reproductive "attractors" of "prehistory".
The term "attractor" refers to the "qualitative" or topological
properties of the solutions of certain non-linear differential equation
systems as represented in the state space of such a
system. An attractor is the limiting behavior of
the system, which many different
trajectories, representing different "starting
points" (birth-states) of the same essential system -- different "initial conditions" -- converge toward increasingly
as the development of the system continues from those initial
conditions in time. The attractor thus represents the "essence" of the
system, which eventually breaks through to appearance from beneath the
distorting layers of initial environment and circumstances if
the self-development of the system continues. It is so named because the
particular closed trajectory of states which represents it in the
state-space appears to "attract" all trajectories in its vicinity into
the vortex of its closed path. An attractor is thus essentially the
state-space representation of a reproduction-cycle, or temporarily
stable evolutionary formation. It might be conceptualized in terms of a
concept of state-space force; a kind of state-gravity, or curvature of the state-space-time
continuum (state-space "field").
Poincaré first discovered attractors in their modern form,
dubbing them 'limit cycles'. Today, they figure prominently, for
example, in
René Thom's "Catastrophe Theory":
"The attractor of a system that is
in dynamical equilibrium consists of the entire
stable cycle of states through which the system
passes.... [S.D.: In highly non-linear systems, a given
attractor may "hold" the system for a very long time, until it suddenly
breaks out of that "rut" and escapes to a higher one]. The
rules governing jumps between cyclic attractors and higher-dimensional
ones are not yet known; they must include not only elementary
catastrophes but also generalized catastrophes, and their study is
today an active area of research in mathematics...." c4
The self-production of man is a process that is punctuated by these epochal stages.
The sentences (1) displayed above are of a peculiar type relative to the forms
of subject-verb-object structured sentences ordinarily encountered at
present. In them, the noun representing the object is the same one as,
or a pronoun referring back to, that representing the subject. The
sentences are thus characterized by a 'subject-object identical'; a unity
of subject and object. In particular, in the above sentences, humanity
is seen to be both the subject and the object of the work of history.
Such sentences have a special name, a
history of their own, long, and strange, and unique properties which
have got them and their formalized analogues a bad reputation in the
annals of formal logic and mathematics. They are called "reflexive sentences" c5
and, when expressed in the ideographic notations of symbolic logic or
mathematical algebras, they are known as "reflexive relations" or
"reflexive functions". c6 Most logicians and mathematicians today would hold that such functions "do not exist" c7, or that they represent paradoxes. c8
That is, we have expressed the fundamental concept of Marxian theory as
a "natural language" version of a "mathematical object" which the
established scientific authorities of our day judge to be nonexistent,
or self-contradictory, and hence, absurd.
It would do us well at this point to examine closely the word "reflexive'. The stem, "flex", means bend, the prefix, "re" applies the connotation of back or again, yielding in totum the sense bent-back. The adjective "reflexive" thus applies to action or process "re-turns"; which curves back upon its origin -- a boomeranging action; a process of feed-back, wherein output is "fed in" again as input.
Now: what has reflexiveness in general, and the reflexiveness of our
fundamental theorem in particular, got to do with dialectics?
(b) Reflexiveness and Dialecticalness
Marx's writing is simply replete with terms denoting such 'self-interactlng' processes, from one end (say, the Theses on Feuerbach & German Ideology) to the other (Capital and the Grundrisse). In the early writings, we encounter phrases like "...self-change... as revolutionary practice" c9 and "mode of self-activity" c10.
In later works we find capital characterized as "self-expanding value",
and the evolution of the human species characterized as
"self-development" c11, and the concept of labour as follows:
"Labour is, in the first place, a process in which both man and Nature participate.... He opposes himself to nature as one of her own forces, setting in motion arms and legs, head and hands, the natural forces of his body, in order to appropriate Nature's productions in a form adapted to his own wants. By thus acting on the external world and changing it, he at the same time changes his own nature. He develops his slumbering powers...." c12
Human praxis for Marx is thus reflexive praxis; capital as exchange-value, accumulated alienated labor-time, is reflexive value -- "value for-itself" c13 or self-related value. The fraction s'/(c+v), the ratio of new value to old, the profit-ratio, is a measure of that self-relation. Reflexive action is conceived as the general 'cause' of evolution, the force of self-development. By 'working back upon themselves' material entities "work out" (elaborate) and real-ize their essence. c14 The concept of dialectic is often associated with the qualities of totality, of contradiction (strife), and of process in the sense of lawful evolution. If we specify dialectical contradiction as internal contradiction -- that is, self-contradiction -- and not simply as conflict between
two mutually external poles, then we have already determined one mood
in which the ordinary conception of dialecticalness is linked to
reflexiveness. But alone, this would constitute but a special case, a
mere intersection. We have also the assertion by formalist
logicians and mathematicians that reflexiveness leads inexorably to
"paradox" and "contradiction" c15,
although it is not immediately evident that these two usages of the
word "contradiction" refer to the same quality.
Nevertheless, a central tenet of this
exposition is that reflexiveness is the very essence of dialectic; the
unity of all its moments. An exploration of the two remaining moments
listed above will likewise converge in the quality of reflexiveness.
Consider the totality. I mean this in its proper sense, that of the full totality, the material universe as a whole, or "Nature". It becomes rapidly evident that, if Nature is a dynamic totality, an evolving process, then it must also be a reflexive totality. By definition, the universe has no others, no out-side. The totality is unique. There is nothing to act upon it externally in order to induce change or motion. c16 If it is developing, it must be developing itself. Its evolution can be accounted for only as the product of its action on itself; its immanent
process -- the interactions among its internal features, that is,
'intra-action', or reflexion. Its motion must be self-induced -- it
must be self-movement. Reflexion, the operation of the totality upon itself, the self-activity of Nature, is both the drive and the expression of its evolution; its evolution's material cause and its material result.
This usage of the term 'reflexion' may come as a surprise to many readers. We are used to thinking of 'subject' as referring only to a human subject, and to hearing 'reflexion' employed to describe an exclusively (in so far as we know) human, mental process, one which does not affect the external matter of the world, the object. That is, we ordinarily class the mental process as intangible, as unmediated by any material agency, (except some still-obscure physiological process in the brain), and as passive, merely mirroring its object without altering object, or mirror, in the process.
But according to Hegel, reflexion is an objective process:
"It is of the utmost importance to know that for Hegel reflection, like all the characters of essence, denotes an objective as well as a subjective movement. Reflection is not primarily the process of thinking but the process of being itself." c17
And, as we have seen above, for Marx, "labour", involving both "imagination" c18 (theory) and praxis, is the 'reflexion' both of humanity upon itself and of Nature upon itself -- the process of self-production. The mental aspect of reflexion is not passive, nor isolated from other modalities of human activity. Rather, it is the inseparable product of human sensuous activity, which changes the world in the process of knowing it, and cannot yet know what it has not yet contacted in the expanding wake of social-reproductive practice:
"...conceive the sensuous world as the total living sensuous activity of the individuals composing it." c19
"The chief defect of all hitherto existing materialism -- that of Feuerbach included -- is that the thing, reality, sensuousness, is conceived only in the form of the object or of contemplation, but not as human sensuous activity, practice, not subjectively." c20
Finally, the mental aspect of reflexion, the contents of the thought-process, the idea itself, is not to be located outside the material universe, as formalists, including mechanistic-atomistic materialists, would have it. According to Bertrand Russell:
"... any statement about the totality (must) fall outside the totality." c21
Rather, the idea is itself a part of the
'historical material'. The concept is also a material object, though of
a special sort, According to one writer:
"The reflective
consciousness was far Marx simultaneously a moment of man's
'practical-critical' activity. The thought always enters into the
reality mirrored by it as an essential component." c22
We find this
interpretation confirmed in Marx's own words. Marx's writings are
studded with turns of discourse which can only count as complete
paradoxes for the object-fetishistic brand of "materialism":
"The development of fixed capital indicates to what degree general social knowledge has become a direct force of production...." c23
"The development of science alone
-- i.e. of the most solid form of wealth, both its product and producer
-- was sufficient to dissolve these communities. But the development of science,
this ideal and at the same time practical wealth, is only one aspect,
one form, in which the development of the human productive forces, i.e.
of wealth, appears." c24
"Regarded materially, wealth consists only in the manifold variety of needs...." c25
"The recognition [S.D.: by the proletariat] of the products as its
own, and the judgment that its separation from the conditions of its
realization is improper -- forcibly imposed -- is an enormous
awareness, itself the product of the mode of production resting on
capital, and as much the knell of its doom as, with the slave's
awareness that he cannot be the property of another, with his
consciousness of himself as a person, the existence of slavery becomes
a merely artificial, vegetative existence, and ceases to be able to
prevail as the basis of production." c26
Therefore, we see mental "reflection" or
consciousness, as but a special case of the much broader law of motion
of reflexion in general, which pervade all of Nature.
In summary then, we
propose that reflexivity is the essence of dialectic. An evolving
totality can only be a reflexive totality, the self-determining agent
of its own becoming. The lawful process or dynamic of such a totality,
grasped as a process-entity, can only be the product of the
self-action of that totality, its material "reflection" upon itself;
its self-causation and recausation (reproduction). Such an entity is
both the subject and the object of its own development. It initiates
the action evoking development and it receives that action as well. It
mediates its own unfolding. The contradiction or inner struggle of such
a totality -- which is expressed in it as becoming, as change -- can be
grasped adequately only as reflexive contradiction, self-struggle. Nature evolves itself by growing out of itself new appendages which reach around, turn back upon and against it. It grows by confronting and struggling with itself in this way. The "conflict between man and Nature" is such a process; it is really the conflict of Nature with itself. The relation of the entity as subject to itself as object, to express it in its most general form, is this "dialectical contradiction". The dialogue in which dialectics consists is this converse between the subjective aspect and the objective aspect of a single entity, this self-dialogue. The "bildung", the negentropic formation, the informatic building-up of the world is the product of reflexion -- of the work of the world upon itself. Reflexion is a material process". Social consciousness is among the "material conditions".
The sub-formations successively thrown up by Nature in the course of its self-unfolding show a rising intensity of self-activity. This increasing intensity of reflexiveness is the trend of cosmological evolution. c27 The highest and most intensive qualities of reflexiveness known to us, called conscious reflexion or self-consciousness, can only manifest given the long telescopic sequence of the preceding orders of reflexiveness as its material basis. Finally, in the human species, the self-amplifying, ascending trend of intensification breaches the threshold whereafter the "mental" aspect of reflexion can emerge.
In terms of historical precedence, if not, on every score, in terms of conceptual content, we take Hegel's "Being-for-itself" -- meaning self-beholding, self-related, or reflexive being -- as our prime paradigm for the Reflexive Functions which express dialectics. c28
While the statements in the summary paragraph above apply with full force only to the full totality, they still apply, but in a spectrum of diminishing degree, to the various sub-totalities or "sub-systems" into which Nature can, with appropriateness, be conceptually discriminated. With sub-systems, which have environments, external determination enters the picture alongside self-determination, or, put another way, their re-flexlon is conjoined with their "flexion" by other systems in determining their phenomenologies.
(c) On the Special Notation Embedded in Marx's Writings
Dialectic, or reflexion, could not be more central to Marx's major theoretical project, the analysis of "Capital". It has everything to do with what this entity "capital" is according to Marx; with the surprising and unfamiliar way he employs this key term -- a way differing radically from that of the political economists, Marx repeatedly characterizes capital as "self-expanding value". This phrase is just the nounal form of a reflexive function, the 'accumulation function' we might call it :
Capital makes itself.
Value produces (more) value.
(reflexive function set 2)
Capital is not a 'thing' in Marx's concept. Its reality cannot be captured in any three-dimensional object, or even in a collection of such. Capital is not a machine, a commodity, a sum of money, a stock certificate, or even the totality of such objects at any one moment, "in-themselves", although each of these is a determined 'particular' of capital, a 'moment' of its existence while it does exist. Rather, capital is a movement, a form of the human life-process or of social praxis, a process mediated through all the "things" named above. It is the movement of self-expansion of
a definite social relationship, called "value"; it is the dynamic of a
determinate historical "totality of social relations of production". c29
And a "totality of social relations of production" is another name for a "society" -- an "ensemble of social relations". Thus, "Capital" is synonymous with capitalist society as a whole. Capital is a historically specific and transient law, or form, of social reproduction; a determinate stage in the self-production of humanity. Thus, the term really denotes the entire movement connecting feudalism (and other use-value dominated social formations) and Socialism. c30 Capital is a social name of a prehistoric form of humanity; the unhuman face of an alienated form of human collective subjecthood. "It" is us.
To do justice to this conception, the (reflexive function set 2) above must be reformulated as follows:
Capital makes more, then less, capital.
Capital makes, and unmakes, itself.
Capital accumulation speeds itself up, then slows itself down.
Capital both posits and negates itself.
Capital causesc31, then discauses, capital.
(reflexive function set 3)
Observe that the
above express, albeit in the barest outline, the complete story of the
historical motion of capital, the historical description or
longitudinal section of capitalism according to Marx. In fact, what the
above formulate is nothing less than the Marxian law of value in its
historical, diachronic view. Especially the third variant brings this
out. It comprises the theory of surplus value, the law-of-value
lawfulness of accelerating accumulation, and the tendency of the rate
of profit to fall, which rate is ultimately reflected in the rate of
accumulation since the former, measured 
, is the upper bound of
the latter; the rate of reinvestment. In so doing, it covers both
the ascendant and the decadent sides of capitalism, and their relation
to one-another -- the period during which the capital-relation promotes
passing into the period during which it enfetters the growth of the
use-value-productive forces.
In the above, we have to fight the language in order to express our
meaning. The display of five variants is not for the sake of prolixity.
It is required to bring out the full connotation of a single
conception, or reflexive function, because of the resistance to such
expression offered by the usage patterns and habitual structures of
modern English. English, at least in its contemporary form, is not
well-crafted for expressing
reflexive sentences, totality-relations, or process-conceptions. It is
ill-suited to speak dialectics. The unconventionality and awkwardness
of the above formulations, their "funny sounding" quality; the
difficulty and unusualness of making 'to cause' a reflexive verb
("self-causes"), and of forming its antonym ('self-discause';
'unproduce', etc.), are all symptomatic of this.
Modern English is the repository, the record -- a reflexion -- of the
life and thought activity of the millions of the English speaking
peoples over the past 200+ years of the unfolding capitalist experience
of humanity -- the experience of capitalist social practice and of the
undialectical, atomistic thought and identity processes which congrue
to that experience and practice. And, as we shall see, the social language of every day
life is the trunk of the tree from which the special languages of mathematics, symbolic logic, and the sciences generally, spring.
The formulations above also bring out the often mystified or disputed meaning of "dialectical contradiction". Not only does this term connote internal or self opposition, but existential contradiction as well. We might replace the compound and contrary verbs in the above -- 'makes more, then less'; 'makes and unmakes', etc. -- with the single verb contradicts:
Capital contradicts Capital.
(reflexive function 4)
"Contradicts" in this sentence is not meant in the "timeless" formal sense, abstracted from the historical continuum, as might result from the appropriate application
of a logical operator which "acts" only on the semantic content of a
truth-valued symbol ( X 
-X ). Here it has a temporal, material,
existential sense. We mean that Capital perpetuates its existence through a process which has as its (later, temporal) consequence the inexistence of Capital: 'existence inexistence'.
The same process which sustains and accelerates new capital formation -- the growth of the productive forces, augmenting reinvestable relative surplus value -- also simultaneously unmakes previously formed capital value, thus eating into the rate of profit, when losses on account of technodepreciated fixed capital are debited to the profit-loss account, hence also curtailing the rate of accumulation, in terms of the value of the total stock of fixed capital, old and new. This has the effect, once the capital-devaluing side of the process gains the ascendancy, of turning capital against the growth of the productive forces -- that is, against itself; its own necessary
tendency and basis -- inaugurating the decadent phase of capitalism,
and finally bringing capital accumulation, the very existence of
capital, to an end, either in Socialism (the only possible form of
renewed growth of the productive forces), or in Barbarism. the unmaking of humanity. Both the 'anabolic' (capital-forming) and the 'catabolic' (capital-destroying) processes
of the capital-value accumulation process; both (1) the rising rates of
relative surplus value, lifeblood of competition and accumulation, and
(2) the devaluation or
obsolescence-depreciation of fixed capital -- the profit-felling,
disaccumulationist, and fictitious-capital c32 generating process, are driven by one and
the same fundamental movement -- the development of the
self-productivity or creative power of the species ("wealth"). When
circulating capital value (value of current product) outweighs fixed
capital value in the balance of the total social capital, the anabolic
process dominates the catabolic one, with the effect of a net positive
rate of exchange-value production and of return on investment (profit).
Capitalism is at its healthiest. It favors and rewards the growth of
the productive forces. But later, with further growth of the
productive forces, objectified as increased fixed capital composition
of total capital, increased "capital intensity" of production, i.e.,
with further accumulation of capital in its veritable "accumulation
form", which is that of fixed capital, the fixed capital value comes to
overbalance the circulating. Then, the catabolism begins to outweigh
the anabolism of value, and capitalism becomes terminally ill. The
rate of profit falls; severe crises explode as the fictitious capital
accumulations, which hide this fall, become suddenly, manifestly
illiquid. Capital has turned increasingly to imperialism abroad,
austerity at home -- looting of the productive forces, nonreproduction
-- in order to supply the "missing" profits and liquidity. To maintain
the rate of profit, Capital's life-principle and prime
measure of success, its personifiers move in every way to transform the fall in the
rate of profit into a fall in the rate of social reproduction; a
growing failure of social-reproductive praxis and management. This
rapidly eventuates in an unviable state of global social reproduction;
a crisis in the self-production of humanity. In this crisis -- one way
or another -- capital must cease to exist.
This is the classical example of the dialectic of self-negation in the
Marxian tradition. We may take its pattern as an archetype of
dialectical or existential contradiction, which process we will meet
many times in the course of this exposition in other realms of cosmic
evolution. A singular content -- in this example, the development of
the productive forces -- both perpetuates and terminates its own outer
complexion, here capital, in the self-coherent, apparently
straight-forward trajectory of its own continuation. We have a
self-curving, self-discontinuing continuum. A single process here has
dual, mutually antagonistic, consequents.
So far, we have used only universal dialectical verbs in the reflexive functions formulated: society produces society, humanity contradicts humanity
; Nature develops Nature, and so on. Suppose we wish to do justice, in
our formulas, to the particular, determinate content and quality of the
self-activity in each case. For instance, though the self-function of
capital falls under the general concept connoted by the dialectical
verbs or relations 'self-produces' and 'self-contradicts', it has
unique, specific features which precisely distinguish its particular
essence. How do we construct a sentence-model of capital such as will
adequately reflect this specificity? One readily available tactic would
be to form the verb or relation-denoting term from the noun, the
subject-object-denoting symbol, itself. This would imply that the mode
of (self)interaction of the subject-object, here Capital, is taken to
be characteristic of that subject-object itself, i.e., a proper part of
the "description" and system of properties which its name, "Capital", represents. For the sentence "Capital self-contradicts", we might
substitute:
Capital 'capitalates'.
Capital 'capitalates' Capital.
(reflexive function set 5)
We
have formed a new, specific verb to
replace by a single word, a more elaborate description of the features
of the capital-process, some of which were just laid out in the
paragraph above. We could go still further in this direction, by merely
assuming that the description of the behavior of Capital, here denoted by
the verb 'capitalates', is absorbed into or properly included by the
description denoted with the noun 'Capital'. Then, we would arrive
at the total disappearance of the separate verb:
Capital (Capital)
"Capital of Capital"
Capital × Capital
(reflexive function set 6)
Where, in effect, we are treating the term 'Capital' as an
operator-symbol, by analogy with an important tendency emerging within
modern mathematics. The operator-symbol is depicted as a function
"of" or as "multiplied against" (applied to) an operand-symbol. Only, in our example, the operand is the same as the
operator. Thus, the expressions (reflexive function set 6) denote the self-operation of
capital. If we "algebraically" replace 'Capital' by 'K', then we can
translate (reflexive function set 6) to K(K) = "K of K" = KK = K·K = K2. The significance of
this tendency to disappearance of the separate verb, and of the
separate "multiplication" operation sign; the implications of this
operator interpretation of self-functions, and of the notational
mapping of reflexion, or self-operation, by self-multiplication, or the "powers" of an operator
symbol, will emerge throughout the entire
course of this exposition.
But what I want to emphasize at this time by this procedure pertains
still to the evolution of "natural" language, not yet to that of the
special language of mathematics. Something is happening, under the
whole trend of our argument, the whole process of inquiry to which we
have been subjecting dialectical or reflexive processes, to the status
of the habitual sentence-structure of our language. That standard
grammatical organization -- reflecting also a deeply ingrained
ontological division at the world-model -- is tending to dissolve under
our analysis! The tripartite division of the world-process into
subject, verb, and object was already partially undermined at the first
onset of explicitly dialectical expression, with our introduction of
the reflexive sentence. The radical distinction between subject and
object was dissolved via their necessary representation by a single
name in sentences expressive of dialectical process. Now, further
along, the verb has come under attack as an ontologically separate
category. It is revealed to be but a moment of the same symbol which
already denotes both subject and object.
We seem headed toward the conception of a single entity, positing
itself in relatively distinct subject, verb, and object moments, but
which is fundamentally one; a unity of these phases or moments. We seem
headed also toward the design of a language suitable to express such a
conception. The self-differentiation of this unity is now expressed
notationally (phonetically) by three words, variants of a single stem,
but with slightly differing suffix (might also be prefix) formations,
reminiscent of the tense and case provisions of a single infinitive
or noun in many languages. (In the examples presented so far, the
"subject" and "object" formations happened to have been identical to
each other and to the "stem").
The verb form is semantically appropriate despite its syntactic
awkwardness. Capital behaves in a definite, lawful way which expresses
its essence; defines it. We have merely reconstructed the verb to
reflect this. That this reconstruction does not come easy to the
language, is not systematically provided for, forcing a break with
conventional modes, testifies rather to the semantic inefficiency of
contemporary English for dialectical purposes; the inbred resistance of
this language as a medium for dialectical thought. But this resistance
is to be expected -- nay, predicted -- on the basis of a Marxian theory
of the connexion of linguistic evolution to social praxis.
The point is
that, for dialectical thought, there is nothing static or inert about
"substance": what an entity is can not be fundamentally
distinguished from what it does:
"As individuals express, their life, so they are." c33
Or, in Hegelian language, essence is the
totality of appearance.
Abundant historical and ethnographic evidence
verifies that the basic Indo-European sentence-structure -- plus the
metaphysics or "ontologic assumptions" implicit in it -- are far from
being the only philosophical groundwork of which the social mind is
capable, or the only grammatical kernel around which a socially-viable
language can be woven. Benjamin Lee Whorf documents structural features of the languages of certain tribal peoples which embody
principles of conception strikingly similar to those evoked above:
"In English we
divide most of our words into two classes, which have different
grammatical and logical properties. Class 1 we call nouns,
e.g., 'house', 'man'; Class 2, verbs, e.g.,
'hit', 'run'.... In Nootka, a language of
Vancouver Island, all words seem, to us, to be verbs, but really there
are no classes 1 and 2; we have, as it were, a monistic view of nature
that gives us only one class of. words for all kinds of events. 'A
house occurs' or 'it houses' is the way of saying 'house' exactly
like 'a flame occurs' or ' it burns'." c34
"Nootka has no parts of
speech; the simplest utterance is a sentence, treating of some event or
event complex. Long sentences are sentences of sentences (complex
sentences), not just sentences of words." c35
Whorf goes on to trace the thread of
subject-verb-object metaphysics from "natural" language all the way
into the heart of the logic and mathematics we have
produced. Far from being simply embodiments of universal, "self-evident
laws of thought", our present mathematics and "symbolic logic" reflect
the ideological presuppositions imbedded in our linguistic practices,
and the thought-practices modulated by them:
"The Indo-European languages and many
others give great prominence to a type of sentence having two parts...
substantives and verbs -- which those languages
treat differently in grammar .... the Greeks, especially
Aristotle, built up this contrast and made it a law of reason. Since
then, the contrast has been stated in logic in many different ways:
subject and predicate; actor and action; things and relations between
things; objects and their attributes; quantities and operations. And,
pursuant again to grammar, the notion became ingrained that one of
these classes of entities can exist in its own right but
that the verb class cannot exist without an entity of the other class,
the "thing" class, as a peg to hang on. "Embodiment is necessary", the
watchword of this ideology, is seldom STRONGLY questioned. Yet the
whole trend of modern physics, with its emphasis on "the field", is an
implicit questioning of this ideology. This
contrast crops up in mathematics as two kinds of symbols
-- the kind like 1, 2, 3, x, y, z and the kind like
+, -, ÷, 
, log-, though, in view of 0, ½, ¾, 
, and
others, perhaps no strict two-group classification holds. The two-group notion, however, is always present at the back of the
thinking, although not overtly expressed." c36
The whole trend of the
Hegelian dialectical critique (and, in fact, of German classical
idealism as a whole) can be mapped as a conscious reflection upon, and
an event of, linguistic evolution; can be projected onto the screen of
the basic Indo-European sentence-form and portrayed there as a critique
and transformation of that form, headed in the direction of a new basic
grammatical paradigm, and a new kind of language. This philosophical
movement, one having the profoundest roots in ongoing
psycho-sociological events, in social praxis -- in short, in history --
was driven by a single impetus: the systematic doubting that object is
absolutely other than subject, and that verb or relation is
fundamentally dirempt from subject-object.
Bertell Ollman has discerned
the actual employment of the beginnings of such a new language in
Marx's writings, together with something of the ontological presuppositions upon which it rests:
"Marx conceives of things as Relations. " c37
"No one would deny that
things appear and function as they do because of their spatio-temporal
ties with other things.... To conceive of things as relations is simply
to interiorize this interdependence -- as we have seen Marx do with
social factors -- in the thing itself. Thus, the book before me
expresses and therefore, on this model, relationally contains everything from the fact that there is
a light on in my room to the social practices and institutions of my
society which made this particular work possible. The conditions of
its existence are taken to be part of what it is, and indicated by the
fact that it is just this and nothing else. In the history of
ideas... this view is generally referred to as the philosophy of
internal relations." c38
Such a mode of thought would immediately
undermine the fundamental distinction upon which, for instance,
modern "symbolic logic" is based -- that between concepts, predicates,
or relations -- usually denoted by upper-case letters -- and
"individuals", "object", or "elements" -- usually denoted by lower case
letters. Relations would be objects and objects relations. This would
make relations (as predicates) predicable of relations (as objects), up
to and including cases predicable of themselves. The tremendous
misgivings and suspicions such a notion arouses among the main-stream
of modern logicians and mathematicians -- even in the minimal form in
which it enters perforce into elementary formal logic, that of the
identity-relation -- is typified by the following quote from Bertrand
Russell. This particular kind of "thing as Relation", which we can call
a "self-Relation", is regarded as bordering on the paradoxical (at
best) are all. Such concepts are all but beyond the ken of the logics which are all we possess at present in developed form:
"There is a certain
temptation to affirm that no term can be related to itself; and there
is a still stronger temptation to affirm that, if a term can be related
to itself, the relation must be symmetrical, i.e., identical with its
converse. But both these temptations must be resisted. In the first
place, if no term were related to itself, we should never be able to
assert self-identity, since this is plainly a relation. But since there
is such a notion as identity, and since it seems undeniable that every
term is identical with itself, we must allow that a term may be related
to itself. Identity, however, is still a symmetrical relation, and my be
admitted without any great qualms. The matter becomes far worse when we
have to admit not-symmetrical relations of terms to themselves.... What
we have to consider is, then, the fact that a predicate may be predicable of itself. " c39
These "Relations" -- including especially
'self-predicates' or 'self-Relations' -- which are the "things", the
units of reality for Marx -- what is their exact nature? The Marxian
usage of the term "Capital" or "Capital-in-General" c40
is a case in point. Denoting that singular entity into which subject,
verb, and object of the capital function are merged, each being
revealed as a mere talification of that single, it is a prime example
of the kind of "symbol" we are driving at; one which denotes such a reflexive entity: a self-process or "being-for-itself". Such terms
evidently do not refer to objects conceived timelessly, or as simply immersed in time, like buoys bobbing in a separate flowing medium. We
are not talking about three-dimensional, spatial entities, as in the
familiar pattern of English thought and expression. Rather, such usages
depict entities as processes or events ('eventities'); as inherently
active substances into which a temporal content enters as an essential
ingredient. Entities are conceived here as being part of concrete
duration -- part of the very fabric of the space-time continuum. Such
'eventities' have a finite, lawfully determinate 'length' in time too,
as well as in the three mutually perpendicular spatial dimensions. That
is, Marx's key terms seem to refer to four-dimensional solids, not to
three-dlmensional ones; to space-time objects, i.e., which are not
merely spatial, but also temporal wholes. c41
Ollman brings out this ingredience of the
temporal in Marx's conception of object-ivity in these words:
"To introduce the temporal
dimension into the foregoing analysis, we need only view each social
factor as internally related to its own past and future forms, as well as to the past and future forms of surrounding factors. Capital, for Marx, is what capital is, was, and will be.... In short, development -- no matter how much face-lifting occurs -- is taken as an attribute of whatever undergoes development." c42
The successive historical transformations
that the capitalist system exhibits strung out along time are conceived
by Marx as all contained in, internal to, part of that entity which Marx
denotes by the symbol "Capital". This entity is, in turn, but a
temporal section of the historical continuum, a portion of the
four-dimensional body of humanity itself.
This novel, Nootkaesque
logico-linguistic principle allows a radical clarification of the
concept of "law" as applied to evolutionary processes. Many people have
been bothered by the fetishistic appearance of figures of speech like
"this law governs such-and-such..." which portray a law -- apparently
nothing but a mental abstraction of the patterns in the flux of
experience -- as a pseudo-subject; i.e., as if "law" operated as an
objective agency out in the world of phenomena. And yet, such reified
figures persist with a peculiar resiliency, even in writers as aware
of the phenomena of alienation and reification as Marx. Where, then, is
a "law" located -- inside our heads or out in the "external" world? Or both? Or neither? Marx's
mode of conception, which takes 'objects' as 4-dimensional totalities,
event-entities, reveals the law of development of such entities to be a
matter of four-dimensional solid geometry. Their law describes their
temporal outlines; the shapes of their time side-views.
"Law" describes, on this view, not some pattern external to an entity
which it "goes through", as three-dimensional thinking would suggest,
but
rather something which is part of the very substance, the
physique, the contents of its being -- the necessary morphology of its
endurance, its continuum; the silhouette of its duration.
The consequences of an action/event/process/object are not, on this
view, separate from it, outside it, beyond it. Strictly speaking, they
are not other than it at all, but belong to its material essence, its
immanent logos, its very materiality. The "effects" which it "causes"
are in fact located within the boundaries of its 4-d (quartic)
volume. Its consequences are thus parts, limbs, organs of its 4-d body.
It is 'composed' of its consequences (as well as of its origins,
causes, premises).
Prediction about the future of such an 'object', once its essence (law)
is comprehended, becomes a matter of geometrical inference as, in 3-d
mechanical drawing, the shape of any one face of an object implies
overall necessary features of its other faces. By looking at one end of
an object, you can infer much about the other end; its general
appearance is implied. Similarly, in 4-d 'Organical drawing' -- the
'drawing' of predictions ("projections") -- by observing the beginning
of an eventity, we can infer much about its end. This notion is
implicit in the oft-mentioned helical model for the dialectical
movement 'negation of the negation', wherein the original state
returns in the end, but at a higher level -- a return with development
(example: 'primitive communism class societies full communism').
c43 Thus, once the essence or "law of motion" c44 of Capital, or Capitalist society, is grasped through the
experience and intensive scrutiny of its 'close' side, its present
(and/or past) face(s), we can ask "what must the far, future end of
Capitalism look like" and get definite answers as to the general
appearance and quality of that future experience.
I don't mean to suggest by this that the temporal 'other end' of the 4-d entity is "already there", or that its lawful unfolding could not be
intercepted, truncated, by supervening events obstructing its temporal
prolongation. I mean only that its future will have a certain necessary
"consistency" and "continuity" with its past and present, a certain
forseeably curved continuum if it does continue its unfolding; and that
this consistency constitutes the very homogeneity of its substance.
That is the meaning of a dialectical law; a law of development.
Furthermore, the 4th length, the finite time dimension, or duration, of
an eventity -- how 'long' it lasts -- is a lawfully
determinate, 'geometric' aspect of its total size, a magnitude
inherent in its geometry, just as its other 3 finite spatial
measurements which express its shape are characterisitic of its nature.
Take a redwood tree. Both its spatial and its temporal "size" are
characteristic of its kind. This is so because real entities are
self-terminating; self-closing in time as well as space. They are
existentially self-contradictory. Their demise comes about even in the
absence of external "accidents", 'interceptions', for internal
reasons. Their dissolution is self-caused, an expression of
the internal limitations of their "form" just as their spatial boundaries
express their interior structure and nature. The boundary, showing
the shape or "form", is the place where the volume or content comes
to an end. According to Hegel:
"When we say of things that
they are finite, we mean thereby... that Not-Being constitutes
their nature and their Being. Finite things are, but their relation to
themselves is that they are related to themselves as something
negative, and in this self-relation send themselves on beyond
themselves and their Being. They are, but the truth of this Being is
their end. The finite does not only change,... it perishes; and its
perishing is not merely contingent, so that it could be without
perishing. It is rather the very being of finite things that they
contain the seeds of perishing as their own
Being-in-self, and the hour of their birth is the hour of their death." c45
According to Marx:
"Our method indicates the
points where historical investigation must enter in, or where bourgeois
economy as a merely historical form of the production process points
beyond itself to earlier historical modes of production. In order to
develop the laws of the bourgeois economy, therefore, it is not
necessary to write the real history of the relations of production. But
the correct observation and deduction of these laws, as having
themselves become in history, always leads to primary equations -- like
the empirical numbers, e.g., in natural science -- which point towards
the past lying behind this system. These indications, together with a
correct grasp of the present, then also offer the key to the
understanding of the past. This correct view likewise leads to points
at which the suspension of the present form of production relations gives signs of its becoming -- foreshadowings of the future. Just as, on
one side, the pre-bourgeois phases appear as merely historical, i.e.,
suspended presuppositions, so do the contemporary conditions of
production likewise appear as engaged in suspending themselves and
hence in positing the historic presuppositions for a new state of society." c46
To summarize: dialectical,
existential contradiction means that any finite being, any object qua
object, makes itself, its own existence, impossible after a definite
time. Any object is thus also the subject enacting its own dissolution,
undoing itself.
It appears to be a basic law of all change, hence a law of being, that
objects, which are finite in each of their three spatial extensions,
will be finite in their fourth, temporal extension as well; will "come
to an end" in time as well as space. We don't ordinarily experience
"everlasting" 3-dimensionally finite objects. Even protons and
electrons should be no more everlasting than the universe itself, at
most, according to current theories, but there is question whether
these phenomena are really c47
"objects" at all. The spatial and temporal finitudes of a given "object" are causally coupled and interdeterminate; forming a 'unity', an
overall quality of finitude characterizing the particular object -- an
overall sense of its 4-d "size", its relative 'magnitude' . This
law appears to be applicable even to the cosmos as a whole. c48
The regularities observed in the spatial characteristics of bodies are
analogous to those observed in the temporal, both in regard to their
'magnitudes' and their shapes or contents (laws). The time dimension
'outline' is subject to "variation" in shape over the multiplicity of
objects of a given kind, just as are the shapes along each of the three
spatial dimensions. However, it must be said that the time-size and
event-outline or morphology is subject to an additional 'uncertainty'
for which the other three admit of no commensurable analogue, but from
which their variations, rather, derive. This uncertainty is associated
with concepts of "degrees of freedom", "free will", and apparent
"chance", "contingency", or "spontaneity" -- with what we will call
"the freedom of necessity", or the creative aspect of becoming and its
laws, the aspect which produces variation, variety, bifurcation. The
greater the degree of evolutionary reflexiveness of the system in
question, the more intensely does it manifest this quality of "freedom"
or "self-creativity".
Nor are "three-dimensional objects" real as such. No entity has finite
spatial extension while being infinitesimal or 'point-thin' in the time
direction. We do not experience such
'instantaneous objects', nor would such an experience be intelligible.
Neither do we experience ordinary objects -- cartons of milk, salt
shakers, and plates -- flashing suddenly in and out of existence on the
kitchen table! There do exist objects of characteristically short
duration -- also usually of characteristically minute spatial extension
-- such as "hyperons", "mesons", and other "unstable" subatomic
"particles". A 3-dimensional time-slice of an object-process is not
self-subsistent as such. It cannot subsist apart from the rest of
itself. And yet formal logic abstracts such slices; treats of objects
as if they were merely spatial and 3-d, and sets these fictions
identical to themselves. That is, fundamental logical laws, like the
law of self-identity -- "for every A, A 
A" -- get into trouble as
models of actuality precisely because "A" here denotes a 3-dimensional
"moment" of what is in actuality a 4-dimensional solid. The conception
of 4-dimensional, self-developing and self-terminating process-objects,
whose "internal consistency" or self-coherence is expressible via
'reflexive functions', demands a new logic, a new kind of notation or
symbolic system to supersede symbols like the "A" above, and a new kind
of mathematics: the mathematics of dialectics. The "internal consistency" just mentioned is fundamentally
characterized by a quality of "self-consistent self-contradiction".
The historical self-development of what Marx calls the "Law of Value"
is the classic example of such "self-contradictory self-consistency". c49
All
of the foregoing explains a good deal of the enormous of difficulty one
hears complained about by new readers of
Marx (and Hegel!), the consistent objections to Marx's language and
"style", and the reason we have undertaken such a prolonged exposition
concerning said style and language. The language in which Marx's
critique of Capital is written is NOT language as we commonly know it!
And the thought-processes which it records and to which it refers are
foreign to those formalistic and fetishistic habits which the lived
illusions and reifications of exchange-value dominated daily life
conditions. Marx's language is rather a special idiom, a dialectical
dialect expressed through ordinary phonetic notation, a
"foreign" language, alien to the language of alienation, transliterated
into the native symbols. In this
dialect, the key words, such as "Capital", "social production", etc.,
refer to tetra-dimensional solidities; space-time objectivities;
quartic dynamicities or "dynamics". NOT to 3-d 'items' without time or
inner restlessness; not to lifeless, static, or even to merely
externally animated "things". Marx's idiom, especially in its raw
form as reveled in the Grundrisse, is actually
a "natural" language, phonogramic version of a new logico-mathematical
notation;
of a new kind of language, and a 'dialectical algebra'. That is,
implicit in Marx's writing is a new kind of conceptual symbolism,
and its features will serve as one of our guides in the exposition that
follows, aimed at bringing this new symbolic system into articulation
through an immanent critique of its existing linguistic, formal
logical, and formal-mathematical forebears.
We are convinced that to understand Marx's work, to decode his
language, requires a thorough exploration of dialectics, and the
gaining of facility in the new mode of thought which it evokes. For
what dwells behind this new language is nothing short of a new mode of
consciousness and, founding that, a new quality of self-identity --
the first glimpses of the psychology of the socialist individual.
(d) The Method of Immanent Critique
"Reason has always existed, but not always in rational form." (Karl Marx) c50
What, then, is the
relation of this embryonic new logic and mathematics -- this new
scientific language and this new kind of science -- to the existing
logic, the existing mathematics, the existing natural and social
sciences produced so far by capitalist society? Is it a mere
"alternative", just another way of doing the same things? Or is it better? Can
it solve problems, and render accomplishable work, intractable,
impracticable for contemporary mathematico-science? Can it lead to
predictions which are outside the ken of that science, and determine
concepts which are beyond it? What, in short, is the
social reproductive value, the potential productive force of the new
language, the new science, whose trail we have already begun to pick up
even within the dim landscapes of prehistoric, presocialist humanity?
Most of us were brought up inculcated with the belief that science,
just like the capitalist economy, was basically sound, strong, and adequate in
its own terms and by its own standards; basically "prospering" and
successful on its own grounds. And this belief is not easily shaken by
contrary facts and experiences, which for a long time tend simply to
not be seen, though they be "staring us in the face". But any good
Marxian -- anyone having attained a critical awareness of the existence
of ideology, and anyone with a sense of the power and necessity of
dialectics, by what ever name it may be called, to the comprehension
and practical mastery of reality, and knowing also that modern science
explicitly rejects dialectical premises, and lacks dialectical
methods -- could only be lead to the profoundest suspicions that deep
inward difficulties must be present and growing in undialectical
science, by virtue of that rejection and lack alone.
A principal implication of the new "reflexive logic" of dialectics, by
virtue of a pervasive quality of it, characterizeable as 'internalism',
as opposed to the "externalism" of reductionist and mechanistic
materialism, suggests that we should not be forced to impose it upon
the old mathematico-science and logic from without. As, in fact, one
test of the validity of this new logic, we should expect to find it
called for, and even already cropping up within -- in tentative,
ghostly versions -- the old mathematico-science itself. This we should
expect to find to be a result of immanent difficulties, difficulties
which the elaboration of that science should encounter in itself, on
its own home ground, at a certain stage, for no other reason than that
of its anti-dialectical, atomistic foundations.
In fact, the quickest way to clarify and build-up the new organon of
dialectics should be to go within the existing science which denies
it, and to delve so deep as to Penetrate deeper than its own
self-recognized foundations. Thereby, we should seek to locate its
internal contradictions, or self-inadequacies, and their roots
in that foundation, and to come back out through, thence beyond, that conceptual organism; to transcend it, having
started from its innermost depths. We should use its internal troubles
as our guide, looking to resolve these troubles by means which
they themselves, plus the rudiments of dialectics already developed,
suggest, but which the old science denies to itself. The development of
these means should in fact be nothing other than a renewed development of dialectics itself, but one locating itself in the actual texture of historical time; the "agenda" of unsolved
problems already confronting the species.
For example, in the section
immediately following this one, we wish to critique the very notion of
the "Set" as an adequate model of the Concept, but we want this to be
an immanent critique of the set-model, that appropriates whatever grain
of useful and valid conception has been worked up around the set
theory. That is, we do not just throw set theory out, dismissing it out
of hand, on the grounds of its atomistic assumptions, and try to start
elsewhere, from scratch, to build an adequate model. That is the 'Utopian'
method. Instead, we see if the set-model can be 'expanded' or
developed under our repeated reflection, i.e., repeated criticism --
exploded from within. We try, thus, to arrive at a progressively more
adequate model via development of the self-contradictions of set theory
itself.
We propose, then, a two-pronged attack on the terrain of
modern mathematico-science, one which we expect will become a pincer
movement, leading to internal envelopment. (1) We have already located
the "natural" language form of dialectical representation -- in the reflexive sentence form. We will trace the
thread of this kind of model of reality past
ordinary speech into the specialized, "artificial" languages of formal
logic and mathematics, to see how well they are able to handle this
kind of model and the kind of processes which it maps. Secondly, (2)
we will look for the major trouble spots and unsolved problems of
modern mathematico-science.
We fully expect that these two lines of
inquiry will converge -- that dialectical processes and models mark the
major impasses of contemporary knowledge.
Therefore, in what follows,
we will examine contemporary logic, mathematics, and finally, natural
science, respectively, with a view to pinpointing, in their most
universal form, the self-obstructions which currently block their way,
which define the present impasse of human knowledge, and the decadence
of the capitalist sciences. We will take these internal troubles
themselves as the keys and clues telling how to proceed in developing a
dialectical science capable of overcoming the fetters which they
reveal, and to solve the urgent problems of human praxis which they
express in abstract form. Our
focus will be led to concentrate most heavily on
three long standing problems: (1) that of the set-theoretical paradoxes in formal
logic; (2) that of nonlinearity in mathematics, and of the absence of
"closed form" (known function) solutions to nonlinear differential
equations for which unique solutions have, however, been proven to
exist, though no one today knows how to find them or write them down,
and; (3) the problem of self-forces, of self-interacting "particles" and fields, in
quantum mechanics, and of the "divergent" or runaway solutions and
infinities to which their present mathematical expression leads (the
problem of so-called "re-normalization"). We will uncover the unity of
these three problems, and we will explore a certain conjecture as to
the conceptual innovations and method which could lead to their
resolution. We will begin to develop the ideographic representation of
said concept and method, constituting the rudiments or an ideography of
dialectics, that is, a dialectical mathematics, or "symbolic
dialectics".
The material presented in the fourth section, on natural
science, owes its discoverability to a novel re-conceptualization of
the meaning of the concept "Dialectic of Nature", one dissenting
both from that advanced by Engels and from that issuing in rejoinder
from the Frankfurt School and Georg Lukacs, but synthesizing essential
features of both. The research, and the exposition, proceeds under the
rubric of the general model:
'Nature develops itself.',
'Nature contradicts Nature.', or;
'Nature # Nature.'
(reflexive function set 7)
Wherein we use the ideogram '#'
(doubly-slashed equals-sign) to replace the relation name or verb
"contradicts".
Under this rubric, we will explore principally three
subordinate reflexive functions:
'stellar nucleosynthesis # stellar
nucleosynthesis'
(reflexive function 8)
'planetary abiosynthesis # planetary
abiosynthesis'
(reflexive function 9)
'biospheric photosynthesis # biospheric
photosynthesis'
(reflexive function 10)
That is, these reflexive sentences, or
ideographic 'translations' thereof, will serve as our models of the
pre-human self-evolution of Nature. The general concept of dialectical
evolution comprehending both the pre-human and human phases of Nature
is that of a self-producing evolution fomented via the reflexive
activity of a nested succession of relatively self-sustaining
processes; successive self-terminating, self-sublating states of the
concrete space-time continuum. These reflexive states are grasped as
'pre-subjectivities', proto-Subjects whose self-interaction or
reflexion constitutes a 'pre-dialectic' or 'proto-dialectic' -- the
dialectic of Nature as it ensues prior to and directed toward the
production of Man.
Throughout the presentation, we will have at the
back of our mind, coming to the surface at appropriate loci, the urgent
unsolved problems of continued social reproduction at the present stage
which, in part, led us to this study. These include (1) that of
describing fusing plasmas, predicting their stabilities and
instabilities by solving the non-linear partial differential equations
by which they are presently described -- solutions necessary to the
mastery of fusion power; the "bringing down to earth" of solar power; (2)
that of modeling a dynamic global economy, i.e. in which the
"technological ratios" -- representing the "productive forces" or
productivities -- are allowed to evolve, and in which labor-power is
accounted as both a producer of all social products, and as itself a
product of the consumption of said products, and hence as a reflexive entity, a producer and
consumer of itself (the problem of "nonlinear programming"), and; (3) that of the strategy and programmatic
formation of an international revolutionary movement, which, to the
extent it grows successful, must take its own growing existence, its
own activity, its own strategy into account as among the social/"environmental" factors on the basis of which it arrives at its
strategy, and program. The second problem above is central to the planning processes of a
self-planning society, and the third is crucial to the process of
organization necessary to realize that self-planning society in the
first place. The first problem must be solved if social evolution is to
continue beyond the exhaustion of the carboniferous 'yolk of the earth'
on which the embryonic nöosphere still feeds.
The
enquiry here recounted leads to a vastly new perspective on what has
been hidden all along -- there implicitly by virtue of conceptual
coherence -- behind the appearance of mathematics as it took shape
along the tortuous path trod by mankind through its prehistory. The
really basic and apparently simple questions, whose answers are usually
taken for granted as already long settled and well-understood, have to
be asked again and answered anew, questions like "What is
Mathematics?"; "What is Number?"; "What is Logic?"; and finally "What is
Language?". Our enquiry leads to the answer that humanity has so far entertained
ideologies about these subjects; has only just recently begun to
penetrate the essence of Number, Logic, and Language -- an essence
which is only now coming into exposure as our horizon expands at last
to the outer boundaries of our Prehistory, giving us the first glimpses
over the edge. The Dim Ages are coming to an end.
The recurrent unexpectedness of
the most powerful new methods and conceptions in mathematics; the way
so many of its developments have taken us -- its developers - by
surprise; the oft-noted "unreasonable effectiveness" of mathematical
descriptions of nature, as well as the specific loci of the
counter-examples to this, where mathematical descriptions have so far
and for so long failed, should have forewarned us that some thing was
afoot. To name only a few examples, the surprisingness of Euler's
identity ei·pi + 1 = 0, the ubiquity of the number e in general; the
"existence" (conceptual coherence and necessity) of new kinds of
number, uncardinal number, like i; the "strangeness" with which they
struck us, leading to epithets like "imaginary"; the
uncanny properties of Leibnitz's pseudo-ratio dy/dx; the still not fully accounted for c51 scope and power of Heaviside's
Operator Calculus, by which he began an algebraicization of differential equations -- all this
and more should have tipped us off that we really didn't know what we
were working with.
There is every indication that the solutions to the
riddles of Mathematics, Logic, and Language take their own inextricable
places in the overall solution to "The Riddle of
History" c52 of which Marx spoke.
To say that humanity does not yet
understand what mathematics -- which it produces and uses -- is,
says that humanity does not yet understand its own faculties, its own
conceptual powers; the nature of its own processes of consciousness. It
means that humanity does not yet know itself. The socialist
revolution is that ceremony, marking the end of Prehistory, in which humanity introduces itself to itself, an experience propagating in an
earthquake-like shock of recognition that will be felt, reverberating
long in after-shock, around the world.
2. Dialectics and the Critique of Formal Logic
no pages currently available
(a) The Internal Contradictions of Set Logic
(b) Leibnitz' Dream (When the Bourgeoisie was Young): Pasigraphy (Logic and Language)
(c) George Boole's Algebra of the Thought Process (Some "Uninterpretable" Propositions Thereof
(d) "Imaginary Roots": G. Spencer Brown's Solution of the Paradozes in "Laws of Form"
3. Dialectics and the Critique of Formal Mathematics
no pages currently available
a) Mathematics Inherits the Paradoxes
b) The Problem of Nonlinearity: The Case of the Missing Functions
c) Nonlinearity, Paradoxicality, and Dialecticality
d) The Fundamental Theorem of Non-linear Algebra?
e) The Theory of Functions of a Hypernumber Variable?
f ) The Prehistory of Mathematics
4. The Dialectic of Nature
no pages currently available
a) Introduction
b) Pre-Atomic Evolution
c) Atomic Evolution
d) Molecular Evolution
e) Cellular Evolution
f ) Metabiotic Evolution
g) Social Evolution
5. Dialectics and Socialism
no pages currently available
a) Dialectical Consciousness and Proto-Socialist Identity
b) Dialectics and the Problem of Socialist Strategy
c) Dialectics and the Problem of Socialist Program
d) Dialectics and the Self-Planning Society
A. Citations
c1 Karl Marx, "Economic-Philosophic Manuscripts of 1844", in T. B.
Bottomore, translator and editor, Early Writings, McGraw-Hill (New York: 1963), page 202.
c2 Karl Marx, "Foundations of the Critique of Political Economy" (hereafter "Grundrisse"), Martin Nicolaus, translator and editor, Penguin
(Baltimore: 1973), pages 540 and 542.
c3 ibid., page
712: "When we consider bourgeois society in the long view and as a
whole, then the final result of the process of social production always
appears as the society itself, i.e., the human being itself in its social relations. Everything
that has a fixed form, such as the product, etc., appears as merely a
moment, a vanishing moment, in this movement. The direct production
process itself here appears only as a moment. The conditions and
objectifications of the process are themselves equally moments of it,
and its only subjects are the individuals, but individuals in mutual
relationships which they equally reproduce and produce anew. The
constant process of their own movement, in which they renew themselves
even as they renew the world of wealth they create."
See also:
- Joseph O' Malley, translator and editor , "Critique of Hegel's
Philosophy of Right", Cambridge University Press (Oxford: 1970),
Editor's Introduction, page xvii.
c4
E. C. Zeeman, "Catastrophe Theory", Scientific American, 234:4, April 1976, page 75.
See also:
- Bremerman, "On the Dynamics and Trajectories of Evolution Processes", op. cit.
c5 The American College Dictionary, C. L. Barnhart, editor-in-chief, Random House (New York: 1963), page 1018, which defines "reflexive" as: "Gram. 1. (of a verb) having identical subject and object, as shave in he shaved himself. 2. (of a pronoun) indicating identity of object with subject, as himself in the example above..." and "reflexion" as: "1. Chiefly Anat. the bending or folding back of a thing upon itself."
c6
In recent years, credit is due chiefly to Lyndon LaRouche and
collaborators in the ICLC for stressing the relationship of "reflexive
functions" to Marxian dialectics.
See for example:
- Lyn Marcus (pseudonym
of Lyndon LaRouche, Jr.), Dialectical Economics, D.C. Heath & Co. (Lexington: 1975), "Foreword", page x.
c7
Ludwig Wittgenstein, Tractatus Logico-Philosophicus, Routledge & Kegan Paul, (London: 1961), sections 3.332-3.333, pages 16-17.
Also see:
- ibid., sections 4.442, 5.251, 5.641, 6.123.
c8 Robert Rosen, "On a Logical Paradox Implicit in the Notion of a Self-Reproducing Automaton", Bulletin of Mathematical Biophysics, 21: 1959, pages 388-389.
c9 Karl Marx, Writings of the Young Marx on Philosophy and Society, Easton and Guddat, Doubleday & Co. (Garden City: 1967), page 401.
c10 Karl Marx and Friedrich Engels, The German Ideology, Progress Publishers (Moscow: 1968), pages 85, 89, 90.
c11 Karl Marx, Grundrisse, in Jack Cohen, translator and editor, Pre-Capitalist Economic Formations, International Publishers (New York: 1965), page 93.
c12 Karl Marx, Capital - Volume I, International Publishers (New York: 1967), page 177. [S.D.: underlines mine]
c13 Karl Marx, Grundrisse, Nicolaus, op. cit., pages 452, 459, 536, etc.
c14 ibid., pages 303, 309, 541.
c15 Bertrand Russell and Alfred North Whitehead, Principia Mathematica (to *56), Cambridge University Press (New York: 1970 , page 61.
c16 Nicholas Georgescu-Roegen, op. cit., page 203.
c17 Herbert Marcuse, Reason and Revolution: Hegel and the Rise of Social Theory, Beacon (Boston: 1960), page 143.
c18 Karl Marx, Capital - Volume I, loc. cit., page 178.
c19 Karl Marx, The German Ideology, op. cit., page 59.
c20 ibid., page 665 ("Theses On Feuerbach").
c21 Bertrand Russell, op. cit., page 38.
c22 Alfred Schmidt, The Concept of Nature in Marx, NLB (Bristol: 1971), page 56.
c23 Karl Marx, Grundrisse, op. cit., page 706.
c24 ibid., pages 540-541.
c25 ibid., page 527.
c26 ibid., page 463.
c27 This proposition, that: 'Reflexiveness produces more reflexiveness', may be called the
'reflexion reflexion', or R2, wherein R = 'Reflexiveness'.
See:
- Pierre Teilhard de Chardin, The Phenomenon of Man, Harper & Row (New York: 1965), page 165 & passim.
c28 Georg Wilhelm Friedrich Hegel, The Phenomenology of Mind, J.B. Baillie, translator, Harper & Row (New York: 1967), pages 304-305.
See also:
- Georg Wilhelm Friedrich Hegel, Philosophy of Nature, A. V. Miller, translator, Oxford University Press (London: 1970), page 303.
c29 Karl Marx, Wage Labour and Capital, International Publishers (New York).
c30 Lyndon LaRouche (using pseudonym "Lyn Marcus"), Dialectical Economics, op. cit., page 133.
c31 Here we must distinguish
"causes" referring to 'self -recausation' or repeated, cycling
self-reproduction, from another term which
would refer to the "original" or "primitive" causation of the system,
here capital, which causes would operate outside and from prior to its existence.
See:
- Capital - Volume IV ("Theories of Surplus Value"), part II, page 118.
c32 Karl Marx, Grundrisse, op. cit., pages 317, 402, 413, 423, 446, 542, 659.
See also:
- Karl Marx,
Capital - Volume III, chapter 25.
- Lyndon LaRouche, Dialectical Economics, op. cit., pages xiv, 8-10, 132, 259-260, 295-300, 368-373, 473.
- Under the name "overcapitalization", see Thorstein Veblen, The Theory of Business Enterprise, Charles Scribner's & Sons (New York: 1904:), pages 229-234.
-
Under the name "redundant financial capital", see A. Milne and J. C.
Laight, The Economics of Inland Transport, Isaac Pitman & Sons
(London: 1965), pages 229-232.
-
Under the name "fictive capital", see Horace H. Robbins, Fictive Capital
and Fictive Profit: The Welfare-Military State, A Political-Economy
Based on Economic Fictions, Philosophical Library (New York: 1974), page
415.
c33
Karl Marx, The German Ideology, op. cit., page 32.
c34
Benjamin Lee Wharf, "Science and Linguistics" in Language, Thought, and Reality,
John B. Carroll, editor, M.I.T. Press (Cambridge: 1956), pages 215-216, cf.: Marx, Grundrisse, op. cit., page 361; Marx, Capital - Volume IV, part 3, page 429. (Theories of Surplus Value).
c35
Benjamin Lee Whorf, "Languages and Logic" in ibid., page 242.
c36 ibid., pages 241-242.
c37 Bertell Ollman, Alienation: Marx's Concept of Man in Capitalist Society, Cambridge University Press (New York: 1971), page 27.
c38 ibid., pages 27-28.
c39
Bertrand Russell, The Principles of Mathematics, W.W. Norton & Company (New' York: 1903), page 96.
c40 Karl Marx, Grundrisse, op. cit., pages 449-450.
c41
ibid., page 258.
See also:
- Ludwig von Bertalanffy, General System Theory, George Braziller (New York: 1968), page 57.
c42 Bertell Ollman, op. cit., page 18.
c43
Jean-Paul Sartre, Search For A Method, Hazel Barnes, translator, Vintage (New York: 1968), page 106;
cf.:
- Michael Lucas, "Guerilla Theatre, the Esthetic, and Technology", Anarchos 3, page 38;
- Andre Gunder Frank, "Functionalism, Dialectics, and Synthetics", Science and Society, Spring 1966, page 146.
- Maxim W. Mikulak, "Cybernetics and Marxism-Leninism", page 149 in Charles R. Dechert, editor, The Social Impact of Cybernetics, Simon & Schuster (New York:
1967);
- Karel Kosik, "The Concrete Totality", Telos 4, Fall 1969, page
42;
- Shoichi Sakata, "The Theory of Elementary Particles and
Philosophy", Lenin and the Development of Science, Education, and Culture, UNESCO Symposium, April 6-10, 1970, Tampere, Finland, translated and presented by Yoichi Fujimoto;
- S.J. Noumoff, "The Dialectic and China", XXth International Congress of Chinese Studies, Prague, Czechoslovakia, August 25-31, 1968;
- Vladimir Nabokov,
Speak Memory, page 204;
- V.I. Lenin, "Philosophical Notebooks", Collected
Works, Volume 38, Foreign Languages Publishers (Moscow: 1963), pages 222, 345, and 363.
- Karl Marx, Capital - Volume I, International
Publishers (New York: 1967), pages 581 and 627;
- Karl Marx, Capital - Volume IV:
"Theories of Surplus Value", Part II, Progress Publishers (Moscow: 1968), page 524;
- Karl Marx, Grundrisse, op. cit., page 266;
- Adolph
Meyer-Abich, "The Historico-Philosophical Background of the Modern Evolution-Biology", E.J. Brill (Leiden: 1964), pages 115-116;
- George Jackson, Blood In My Eye;
- Lyn Marcus, Dialectical Economics, op. cit., page 42;
- Kenneth Boulding, A Primer on Social Dynamics: History as Dialectics and Development, Free Press (New York: 1970 ), page 40.
c44
Karl Marx, Capital I, op. cit., page 10: "It is the ultimate aim of this work, to lay bare the economic law of motion of modern society...".
c45 Herbert Marcuse, Reason and Revolution, op. cit., page 136.
c46 Karl Marx, Grundrisse, op. cit., pages 460-461.
c47 Erwin Shroedinger, "What Is An Elementary Particle?", Publication 4028, The Smithsonian Institution (Washington: 1951), page 183.
c48 Misner, Thorne, and Wheeler, Gravitation, W. H. Freeman & Company (San Francisco: 1973), pages 1196-1203.
c49 Karl Marx, Capital - Volume I, op. cit., pages 583-584.
See also:
- Marx, Grundrisse, op. cit., page 401: "Capital, as the positing of surplus labor,
is equally and in the same moment the positing and the not-positing of
necessary labour; it exists only in so far as necessary labour both
exists and does not exist."
c50 Karl Marx, Letter to Arnold Ruge, September 1843, in Writings of the Young Marx on Philosophy and Society, op. cit., page 213.
c51 Gregers Krabbe, Operational Calculus, Springer-Verlag (New York: 1970), pages vii, 103 (5.42.4), and 104.
c52 Karl Marx, Economic-Philosophic Manuscripts, op. cit., page 155.
B. Annotations
a1
This does not in the least imply, however, that the non-intersection
described above has been a one-sided affair. Dialectics has languished
in dormancy while the concepts and methods which it alone could clarify
and justify were developed in a limping form by the reductionist
tradition of bourgeois science, under pressure of the enormous
expansions of horizon in social praxis, the experience of the species
and its penetration of nature, albeit in capitalist form, during this,
century. Dialectics can come of age again, and become contemporary,
only by assimilating all the wealth of this development by way of the
immanent critique of atomistic science.
C. Graphics Credits
D. Post-Publication Notations
E. References
r1 Gotthard Gunther, "Formal Logic, Totality, and the Superadditive Principle", Biological Computer Laboratory Report No. 3.3, page 1.
See also:
"Time, Timeless Logic, and Self-Referential Systems", Annals of the New York Academy of Sciences, Vol. 138, article 2, February 6, 1967, pages 396-406.
r2 Oskar lange, Wholes and Parts: A General Theory of System Behavior, Pergamon Press (New York: 1965).
r3 H. R. Maturana, F. G. Varella, and R. Uribe, "Autopoiesis: The Organization of Living Systems, its Characterization and a Model", Biosystems 5: 1974, pages 187-196.
See also:
H. Maturana, "Neurophysiology of Cognition", in P. Garvin, editor, Cognition, Spartan Books (New York: 1969), pp. 3-23.
r4 Francesco Varella, "_A Calculus For Self-Reference", International Journal of General Systems, Vol. 2, 1975, pp. 5-24.
r5 Nicholas Georgescu-Roegen, The Entropy Law and the Economic Process, Harvard University Press (Cambridge: 1971).
r6 Michael Kosok, "The Formalization of Hegel's Dialectical Logic", International Philosophical Quarterly, 6:4, 1966, pages 596-631.
r7 G. Spencer Brown, Laws of Form, Bantam (New York: 1972).
r8 Charles Muses, "The First Nondistributive Algebra, with Relations to Optimization and Control Theory", in Functional Analysis and Optimization, E. R. Caianiello, editor, Academic Press (New York: 1966), pages 171-212.
See also:
Charles Muses, "Time, Experience, and Dimensionality: An Introduction To Higher Kinds of Number", Annals of the New York Academy of Sciences, op. cit., pages. 646-660.
Charles Muses, "Hypernumber and Metadimension Theory", Journal for the Study of Consciousness, 1:1, January-June 1968, pages 28-48.
Charles Muses, "The Meaning or Cybernetics in Revolutionary Systems", in Proceedings of the International Congress on Cybernetics, J. Rose, editor, Gordon and Breach (New York: 1970), pages 175-199.
Charles Muses, "Working with the Hypernumber Idea", in Consciousness
and Reality, Charles Muses and Arthur Young, editors, Avon (New York:
1972), pages 448-469.
Charles Muses, "Hypernumbers and Their Spaces: A Summary of New Findings", Journal for the Study of Consciousness, 5:2, 1972-3, pages 251-256.
r9 Lars Lofgren, "An Axiomatic Explanation or Complete Self-Reproduction", Bulletin of Mathematical Biophysics, 30:3, September 1968, pp. 415-425.
r10
Heinz von Foerster, "Logical Structure of Environment and its Internal
Representation", International Design Conference, Aspen, Annual, 1962, pages 29-38.
r11 Hans J. Bremerman, "Transcomputationality", (unpublished).
See also:
Hans J. Bremerman, "On the Dynamics and Trajectories of Evolution Processes", in Biogenesis and Homeostasis, Springer-Verlag, 1971.
Hans J. Bremerman, "Algorithms, Complexity, Transcomputationality, and the Analysis of Systems", Proceedings of the Kybernetik Congress", Nurnberg, March, 1973.
Hans J. Bremerman, "A Universal Topology" (Review of Thom's Book), Science, vol. 181, 10 August 1973, pages 536-538.
r12 René Thom, Structural Stability and Morphogenesis: An Outline of a General Theory of Models, W. A. Benjamin Reading: 1975).
r13 Wilhelm Gottfried Leibnitz, Logical Papers: A Selection, G. H. R. Parkinson, editor and translator, Clarendon Press (Oxford: 1966), pages 10-11 et passim.
r14 George Boole, The Mathematical Analysis or Logics Being An Essay Towards A: Calculus or Deductive Reasoning, Barnes & Noble (New York: 1948).
See also:
George Boole, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Dover (New York: 1958).
r15 Benjamin Lee Whorf, Language, Thought, and Reality, M.I.T. Press (Cambridge: 1956).
F. Publication History
Release
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Date
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Released By
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Format and Features
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v1.0
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September
1976
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Studies in Dialectics
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Paper-published original text written by Sinek Docchi.
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v2.0
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July
2006
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Adventures in Dialectics |
HTML onto the internet; includes new graphics, reviews & corrections by the author. Includes sections 0 and 1.
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v2.1
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August
2006
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Adventures in Dialectics |
Addition of section 2.
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v2.x
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Fall-Winter 2006
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Adventures in Dialectics |
Additions of further text.
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G. Contact Information
