Archive for May, 2009

The Human Spirit’s Pursuit Of Self-Liberation

May 30, 2009

“Meaning becomes the moving force of self-transcendence, which reaches beyond the limited horizon and longs for the whole.” (Erich Jantsch, The Self-Organizing Universe)

If universal structure is co-contemporary with the world and its history then we should be able to find this structure waiting to be discovered. I believe this structure has always existed, unconsciously no doubt, in the historical context of people and culture. Levi-Strauss’s structuralism, especially as it is described in The Savage Mind, is just one attempt at disclosing this structure. The holism/ elementarism debate,–the tension that exists between group demands and individual desires–is another attempt at disclosing this structure. This tension exists in all cultures, but varies in degree. For instance, it appears as though Pre-modern man, in the early stages of his development, was able to maintain cultural stability while at the same time maintaining a holistic perception of his environment. This ability, in the words of the anthropologist Levy-Bruhl, “sets Pre-modern man apart from his modern predecessors.” The qualities that we take for granted, or our ability to differentiate the space that surrounds us ad infinitum, did not exist for Pre-moderns. Rather, his/her experience of the participation process was more restrictive and inclusive within what Levy-Bruhl called the “synthetic whole”. In other words, Pre-modern society was considerably different from modern society. Still, all societies must have some mechanism to preserve and perpetuate the social roles that are vital to the on going existence of the group. For Pre-moderns, as for the rest of us, this mechanism lies in our work. In this respect the investigations of Ernst Cassirer become extremely helpful.

Myth, or the mythical-religious consciousness of man, for Cassirer, is understood to be the proto-reality out of which symbolic forms evolve e.g. language, art, religion, science etc. These symbolic forms, in turn, are thought to result from the human spirit’s progressive movement towards more liberated forms of self-expression. From within the matrix of mythical thought, according to Cassirer, evolves the differentiation of the “I” of our personality and, over time, the more potent symbolic forms that define the present state of our modern knowledge and belief.

The origin of the self-liberation process (and knowledge in general), is first discovered in mythical thought as the capacity to order and differentiate, and then the self liberation process, in its capacity to transcend its own reality, metamorphizes into higher levels of symbolic expression. These higher levels of symbolic expression move self-liberation in the direction toward more constancy, endurance and certainty. Cassirer informs us:

“For a glance at the development of the various symbolic forms shows us that their essential achievement is not that they copy the outward world in the inward world or that they simply project a finished inner world outward, but rather that the two factors of “inside” and “outside,” of “I” and “reality” are determined and delimited from one another only in these symbolic forms and through their mediation. …The crucial achievement of every symbolic form lies precisely in the fact that it does not have the limit between I and reality as preexistent and established for all time but must itself create this limit–and that each fundamental form creates it in a different way.” (Ernst Cassirer, Mythical Thought, 1955, p. 155-156)

Cassirer tells us that Pre-moderns, as they engaged their environment through emotions, desires and work, acquired the ability, via symbolic representation, to objectify nature–the nature of both “inner and outer reality.” There was (and is) a double movement that arises from one’s interaction with his/her environment; in one direction there develops an objectification of one’s self-nature and in the other direction there arises the objectification of the social and cultural contents of society. From Cassirer’s point of view, art, myth, magic and ritual are co-creative products arising from this objectifying movement, which in turn, arises from the work that people do in society. “For the form of society,” Cassirer states, “is not absolutely and immediately given any more than is the objective form of nature, the regularity of our own world of perception. Just as nature comes into being through a theoretical interpretation and elaboration of sensory contents, so to the structure of society is mediated and ideally conditioned reality.” (Ernst Cassirer, The Philosophy of Smbolic Form, 3 vol., vol. 2, Mythical Thought, 1955, pl 193) In his three volume work, The Philosophy of Symbolic Forms, Cassirer concentrates his focus on the nature and origins of symbolic form as it first arises in language and myth and then, over time, develops into the theoretical orientations of scientific thought. The utility of symbolic forms, if that is the right word, is not just about a “thing” to be apprehended, it is about a movement towards constancy, endurance and certainty, and that objective applies to both culture and mind.

Pre-moderns then, as participating agents in an environment conceived holistically, objectify mind and culture in and through creative acts of differentiation. This process evolves out of the acquisition of life’s necessities to the creation of more complex societal structures, e.g., kinship systems, sacred and profane boundaries, talismans, origin myths, etc. Thus, myth, or the mythical-religious consciousness of man, for Cassirer, is understood to be the precursor to the technological culture that, from the standpoint of utility, increases our ability to do work, as is makes life easier for all. However, this is not the end of the story. Self-liberation or the movement towards constancy, endurance and certainty continues to direct the human spirit’s progressive movement towards new forms of self-expression.

For Cassirer, myth and myth making becomes an expression of the human spirit/culture as it seeks to liberate itself from the restrictive conditions that hinder and retard the self-liberation process. In our present modern episteme, as Foucault likes to call it, however, the myth-centered universe of the Middle Ages has given ground to a more matter-centered, self-centered universe, a universe that, for Cassirer, represents a more spiritually liberated state, but, for Foucault, represents just another power/knowledge driven episteme, catering to the needs of those who desire and benefit most from power/knowledge relationships. Shortly, I will challenge this assumption by Foucault, but first I must add a bit more structure to Cassirer’s self-liberation world view. This structure comes from an unlikely source, the philosophy of Jean Paul Sartre.

The Innate Structuring Capacity Of All Structures—Reciprocal Movement

Reciprocal Movement, –The Carrier Of Free Thought, The Same Free Thought That Brings Into Being Language, Myth, Science, Ethics, And Civilization

Identifying Sartre’s philosophy as structuralism is, I am aware, pushing the envelope. However, an authority on structuralism has proposed this option (without, I might add, elaborating on it.) “One might go as far as to say…that structuralism is analogous to Sartre’s view of consciousness — it is what it is not, and it is not what it is.” [Jean-Marie Benoist, A Structural Revolution, 1975, p. 1] In Sartre’s book Being And Nothingness, his chapter on Being-For-Itself is subtitled “Immediate Structures of the For-Itself.” [Jean-Paul Sartre, Being And Nothingness, 1966, p. 119] Structure is not hidden in Sartre; it’s just that on the whole Sartre’s book is a polemic against reading structure as anything more than appearance.

In the representation of Sartre’s though
t as “consciousness is what it is not, and it is not what it is,” we find reciprocal movement, the same reciprocal movement encountered, in one form or another, in all the structuralists I have discussed hitherto in this paper. Specifically, Sartre defines the consciousness of the transcending For-itself (our self-space) as: “Consciousness is a being such that in its being, its being is in question in so far as this being implies a being other than itself.” [Ibid. p. 801] In an extrapolation from Sartre’s definition of the consciousness, Benoist describes that relationship as: “it is what it is not, and it is not what it is,” while I describe it as: being-what-is-not-while-not-being-what-is. In both cases, however, we end up with a definition for reciprocal movement.

This double movement is represented on many levels in Sartre’s exegesis on being and nothingness. This double movement becomes very specific in Sartre’s description of his pre-reflective Cogito. In so far as we find ”nothingness” at the center of Cogito, consciousness per se must be understood to be set apart from itself, therefore, Sartre’s pre-reflective Cogito will always form one pole of our conscious experience while the “objects” of consciousness will take their place at the other pole of conscious experience. In this way, Sartre is able to dispense with Descartes’ Cogito on the grounds that consciousness cannot be separated from its object. This condition, where the pre-reflective Cogito becomes a preexistent condition for the conscious awareness of objects, establishes the double movement of conscious reflection — the object of consciousness less the pre-reflective Cogito, and the pre-reflective Cogito less the object of consciousness. Depending on where “you” focus your concern, the content of consciousness is either pushed to the front of consciousness (the unreflective consciousness), or, the object of consciousness is pushed into the background, as the “negation of consciousness” is brought into the foreground (the reflected upon object of consciousness).

Together, our pre-reflective Cogito and the object of consciousness, form our conscious experience of the knower-known dyad. In so far as this double movement turns on the pivot point of pure negation, the known exists for the knower, but the knower can never be fully known. As self-consciousness rises in consciousness, it is denied the possibility of becoming fully self-aware. This result, the incompleteness of self, brings us back to Sartre’s original definition of consciousness, or, “consciousness is such that in its being its being is in question in so far as this being implies a being other than itself.” This movement, the symbol-generating movement of free thought, the movement that makes thinking possible, emancipates language, myth, science, and morality. In the absence of this movement, “thinking” is restricted to the manipulation of signs—mere sensual indicators, minus the symbols that carry the significance of those same indicators.


The Breakup Of A Universe Of Purpose And Order

May 22, 2009

Classical Episteme Compared To Foucault’s Modern Episteme

If Foucault’s reconstituted historical evaluation sounds pessimistic or even nihilistic, I have succeeded in accurately describing his thought, for he is commonly criticized for his nihilistic views. At this point, however, I would like to be a bit adventurous and interpret Foucault’s power/knowledge relationship in such a way that it reflects Foucault in a positive light, as it becomes, in my reading of its most elemental form, the answer to the questions, “Who participates?” and “Wherein does this participation take place?”

Foucault’s analysis of the classical and modern episteme, it seems to me, is a reshaping of the holism/elementarism debate. By recognizing that culture generates from power/knowledge relationships, Foucault has given this debate its ontological legs, so to speak. According to Foucault, it is power/knowledge relationships (the subjugating and the subjugated) that lies behind both the classical episteme and the modern episteme.

Foucault’s description of the classical episteme is consistent with the way I have described the medieval culture. For Foucault the classical episteme is characterized by representation and resemblance. Life, language and labor, through a discourse with the “other,” participate within a universe of order, unity and purpose. Difference occurring within sameness, I believe, is how Foucault describes medieval culture. Difference, divergence, disjunction, are all affirmed as integrals of the whole. According to Alan Sheridan, “classical representation was a universal, neutral, conscious, ‘objective’ mode of thought in which, though it was operated by and for men, ‘man’ as a concept was absent.” [Alan Sheridan, Michel Foucault, The Will To Truth, 1980, p. 82] In the classical episteme everything that “is” is intelligible because of its very capacity for existence, and, reason, through its ability to represent essences, is the tool that is used to recognize the universal essence of all individuating matter.

During the Renaissance, according to Foucault, the classical episteme began to change into the modern episteme. Knowledge of the world became an interpretation of signs, and difference and sameness became organized around interpretation. In the classical episteme, whereas “the complex doctrine of ‘resemblances’ defined the way in which things were linked to each other,” in the more modern episteme, “the doctrine of ‘signatures’ became concerned with how these marks of resemblance were recognized.”[Mark Cousins, 1984, p. 31] The real break with the classical episteme, however, occurred with the philosophy of Descartes. Foucault describes this break:

“…(W)e must not forget that Descartes wrote “meditations”–and meditations are a practice of the self. But the extraordinary thing in Descartes’s texts is that he succeeded in substituting a subject as founder of practices of knowledge, for a subject constituted through practices of the self.

“….In European culture up to the sixteenth century, the problem remains: What is the work which I must effect upon myself so as to be capable and worthy of acceding to the truth? To put it another way: truth always has a price; no access to truth without ascesis. In Western culture up to the sixteenth century, asceticism and access to truth are always more or less obscurely linked.

“Descartes, I think, broke with this when he said, ‘To accede to truth, it suffices that I be any subject which can see what is evident.’ Evidence is substituted for ascesis at the point where the relationship to the self intersects the relationship to others and the world.” [Paul Rabinow, 1984, p. 371]

Man Becomes The Being In Whose Being Being Comes Into Question

From The Perspective Of The Human Spirit’s Pursuit Of Liberation Things Look Different

In the modern episteme difference asserts its right against sameness in truth. In so far as everything is seen to be carrying within itself some “hidden truth,” the search for truth characterizes this episteme. Since everything can be reduced to its constituent parts an elementaristic attitude prevails in this episteme. Formalization takes the place of representation and man becomes the being in whose being being comes into question. The being of man becomes the “shared being” of questioners. For Foucault, at the crossroads of the classic and modern episteme stands Descartes, but the truly modern break with the classic episteme came with the Kantian perspective on man and nature. In Kant’s philosophy the classical episteme becomes reduced to a mere metaphysic. In The Order of Things Foucault writes:

“The Kantian critique…questions representation, not in accordance with the endless movement that proceeds from the simple element to all its possible combinations, but on the basis of its rightful limits. Thus it sanctions for the first time that event in European culture which coincides with the end of the eighteenth century: the withdrawal of knowledge and thought outside the space of representation. That space is brought into question in its foundation, its origin and its limits: and by this very fact, the unlimited field of representation, which Classical thought had established, which Ideology had attempted to scan in accordance with a step-by-step, discursive, scientific method, now appears as a metaphysics. [Alan Sheridan, The Will To Truth, 1980, p. 68]

As is readily apparent, on Foucault’s reading, the classical episteme provided no place for man to engage in his “search for man.” In the classical episteme truth was an already determined relationship. It was, in its multivariate diversity, one’s relationship to God: “In the classical age the truth of a thing was defined by its position in the table of representations, which was constructed to mirror God’s Order–itself an order of the visible.” [Romand Coles, 1992, p. 67] This order was not one of a Godless world in which man is the sovereign subject for all possible knowledge, nor was it a signification of the emptiness, the circumference of which, varying in the inverse relation to the depths plummeted by man in his search for meaning. In an organic, holistic universe there was no need for a differentiated locus of knowledge, there was simply “knowledge of the order of things.”

This “knowledge of the order of things,” however, became integrated in people who had economic needs to satisfy and as conditions changed e.g., centralized monarchy, urban expansion, birth of guilds, etc., people created for themselves less restrictive lifestyles. In this process the scientific revolution played a significant role. Eventually, the holistic, God-centered universe of the Middle Ages gave ground to the matter-centered, self-centered universe of today. If we are to understand this event as something more than the latest development of Foucault’s power/knowledge driven modern episteme we must turn our attention to a description of what Ernst Cassirer called the “human spirit’s pursuit of self-liberation.” From this perspective Foucault’s power/knowledge relationships, when considered as part of the human spirit’s pursuit of self-liberation, become no more than the cultural manifestations of the “synchronic movement” that defines structure.

Foucault’s The Power/Knowledge Subjugation Of Discourse

May 16, 2009

Western Man’s Knowledge Is A Product Of Endless Interaction Between Desire And Power

The Individual, For Foucault, Is A “Cultural Ensemble” Where All Thought And Practice Become Mere Contingencies Of The Social Milieu

Since Foucault’s ideas tend to devalue discursive thought, it is understandable that his ideas were not held in high regard by Piaget: “Foucault has it in for man; the human sciences he views as a merely momentary outcome of ‘mutations,’ ‘historical a priorities,’ ‘epistemes’; these follow one another in time, but their sequence has no rationale.” [Ibid. p.129] Foucault represents a real threat to Piaget, as he represents a threat to anyone who buys into the Enlightenment ideas of progress, perfectibility and purposeful rationality. Foucault’s iconoclastic thought focuses on the individual as a “cultural ensemble” where all thought and practice become mere contingencies of the social milieu. For Foucault there is no depth to a person. A person amounts to no more then the surface attributes of the society that defines him/her. Foucault calls his investigations into Western knowledge “archaeology of thought,” and what this archaeology has unearthed is:

“…not man himself, but the systems of rules and systematic distinctions which account for the transition from one episteme to another, i.e. which explain how our culture has changed from one code to another, how certain formal conditions of possible scientific and non-scientific discourses have been replaced gradually by others.” [Jean-Marie Benoist, The Structural Revolution, 1975, p. 18]

Foucault, in an odd sort of a way, could be described as an “organic Kantian.” Foucault, however, replaces Kant’s categories of mind as the determining agent of knowledge with what he calls the “historical a priori”–the conditions of the possibility for knowledge to arise: “This a priori is what, in a given period, delimits in the totality of experience a field of knowledge, defines the mode of being of the objects that appear in that field, provides man’s everyday perception with theoretical powers, and defines the conditions in which he can sustain a discourse about things that is recognized to be true.” [Michel Foucault, The Order of Things: An Archaeology of the Human Sciences, 1973, p. 158] The more common name for the “historical a priori” is episteme, which put more simply is, “the total set of relations that unite, at a given period, the discursive practices that give rise to epistemological figures, sciences and possibly formalized system’s of knowledge.” [John Sturrock, 1979, p. 92] Epistemes, as Piaget has observed, are arbitrary upsurges, impermanent in nature, and place no restraints upon thought. But, they do not arise out of nothing.

By disregarding any pretense toward a self-determining subject, Foucault has demonstrated how it is possible to achieve spectacular results in the analysis of social practices. In his early books of Madness and Civilization (1961), Birth of the Clinic (1963), and The Order of Things (1966), Foucault analyzed psychiatry, medicine, and the human sciences respectively, and concluded that, according to John Sturrock:

“…the distinctions between madness and sanity, sickness and health, and truth and error were always a function of the modality of discourse prevailing in centres of social power at different periods. In Foucault’s view, this modality was in turn less a product of an autonomous exchange between hypothesis and observation, or theory and practice, than the basis of whatever theory and practice prevailed in a given period. And it followed for him that, finally, the modern history of Western man’s ‘will to knowledge’ had been less a progressive development towards ‘enlightenment’ than a product of an endless interaction between desire and power within the system of exclusions which made different kinds of society possible.” [Ibid. p. 91]

Power And Knowledge, In Foucault’s Structuralism, Directly Imply One Another

We come to the crux of Foucault’ analysis of the episteme when we ask the question, “Who does discourse serve? Epistemes, for Foucault, are characterized by power/knowledge relationships that demand that if the individual wants more out of life than pain and suffering then he or she must submit to the powers that be. It is Foucault’s thesis that “this subjugation occurs, without the subject’s knowledge, in the society wide procedures which pin identities to individuals.” [Mark Cousins, Athar Hussain, Michel Foucault, 1984, p. 254]

In the historical record of the exclusion practices put on the insane, criminals, and sexual deviants, we discover the groundwork that would evolve into, not only the compartmentalization of clinics and prisons, but also the strategic and tactical maneuvers that define today’s modern army. Blanchot expands upon this connection, which was first pointed out by Foucault, in his analysis of the plague (Black Death) of 1348. It was, says Blanchot, “…through a strict parceling out of the contaminated space, through the invention of a technology for imposing order that would affect the administration of cities, and through the meticulous inquests which, once the plague had disappeared, would serve to prevent vagrancy (the right to come and go enjoyed by ‘men of little means’) and even to forbid the right to disappear, which is still denied us today, in one form or another,” that ended up in the laws, techniques, and procedures that redirect our attention away from power/knowledge relationships per se, the same power/knowledge relationships that subjugate one discourse over against another. [Michel Foucault, Maurice Blanchot, Foucault/Blanchot, 1987, p.84] In fact, Foucault implores us to recognize power/knowledge relationships for what they really are: “…power produces knowledge (and not simply by encouraging it because it serves power or by applying it because it is useful); that power and knowledge directly imply one another; that there is no power relation without the correlative constitution of a field of knowledge, nor any knowledge that does not presuppose and constitute at the same time power relations.” [Michel Foucault, Discipline & Punish, The Birth Of The Prison, 1979, p. 27]

Mind Is An Extension Of Natural Structure

May 9, 2009

Mind Is An Extension Of Natural Structure For Piaget

The Developmental Stages Of Children And The Accommodation/Assimilation Process That Allows For the Intelligent Navigation Of Our Environment

From his experiments and observations with children Piaget accumulated evidence that tends to support his belief in the mediating and developmental effects of the interaction of a child with his/her environment. Piaget identified three developmental stages that a child must pass through before a child can attain a mature state of psychological development. The first stage, as has already been pointed out, is the sensorimotor stage. In this stage the infant encounters a resistant environment and, in the process, is made aware of objects, space, time, and causality. At this stage the infant acquires what Piaget calls a “practical intelligence.”

In the preoperational or representation stage of development the child evolves toward the possession of concrete operations. At this stage of development, up to and including the elementary years, the child learns the implicit nature of concrete operations, that is, the child no longer has to “go through the movements” in order to coordinate its own activities in a recognizable and ordered world. The term concrete signifies that the matters of fact materials that a child uses to measure his/her world are real “things” e.g., people, physical objects etc. It is only after the child has reached the stage of formal operations, sometime between the ages of twelve and fifteen, that the child is able to conceptualize what is not perceived (e.g., principles of conservation, reversibility, transitivity, etc.) in his/her capacity to invoke reasoned judgments and deductive thought.

Piaget believes that he can specify the logical operations which occur at each of the three developmental stages i.e. sensorimotor, representational, formal operative. To my knowledge the evidence for the validation of this claim is still inconclusive, however, it seems to me that Piaget’s theory is so well articulated that it is only a matter of time before it will be conclusively rejected or confirmed.

Before I move on to Michel Foucault, the last structuralist I want to talk about in this paper, I believe a few words are in order concerning where the “self” is located in Piaget’s constructivist structuralism. In so far as Piaget locates intelligence in structure and understands structure to arise from a constructive process of continual assimilation and accommodation of an organism in its environment, you could say that Piaget was taking a transcendental Kantian perspective on self. But, in so far as Piaget locates structure in nature as much as he locates it in the mind, and in fact understands mind to be an extension of natural structure, he dissociates himself from Kant as he takes a more holistic position on self. This position separates Piaget from the structuralists I have dealt with previously in this paper. This expanded holism, however, does not come without a price.

The person who we identify with as being “the person who we are” becomes a person only in so far as he/she accommodates and assimilates his/her environment. Accommodation is herein understood to be a change in the assimilated product of environmental interaction i.e., acting on the past to create a present. In the same way assimilation is herein understood to be an action actively reproduced in such a way as to incorporate new (accommodated) objects into one’s own assimilated experience i.e., actualizing the potential to intelligently navigate a course through an uncertain future.

Piaget’s Self Is Located In The Interdependence (RM) Of The Activity Of Content/Form

The Continual State Of Structuring Structures

The locus of this “constructionist self “ is called by Piaget the center of functional activity. The center of functional activity is not, according to Piaget, located in the traditional “me space” that we so often take for granted; nor is it located in the “lived space” that is described in the works of various phenomenologists and existentialists; nor is it located in the positivists physico-chemical brain activity, Nietzsche’s will to power, Marx’s economic determinate or Durkheim’s normative order. Rather, Piaget locates the “constructionist self,” in general terms, “somewhere midway between the nervous system and conscious behavior (because) ‘psychology is first of all a biology.’” [Piaget, 1970, p. 138] In more specific terms, Piaget locates the “constructionist self” in the structure of content/ form interdependence. Piaget explains:

“But what manner of existence is left, then, for the mind, if it is neither social, nor mental in the subjective sense, nor organic?
…If it is, as Levi-Strauss says, necessary to “reintegrate content with form,” it is no less essential to recall that neither forms nor contents exist per se: in nature as in mathematics every form is content for “higher” forms and every content form of what it “contains.”’ [Piaget, 1970, p. 112]

We see in this “double movement” of form and content the real location of Piaget’s self. In this situation it becomes tempting to identify Piaget’s self as the structure of structures but this conclusion is prohibited, as Piaget points out, by Godel’s proof that the ideal of a structure of all structures is unrealizable. Therefore, Piaget retreats into the only description of self that is left, given the limiting conditions set down in his constructionist philosophy:

“…(T)he subject’s activity calls for a continual “de-centering” without which he cannot become free from his spontaneous intellectual egocentricity. This “de-centering” makes the subject enter upon, not so much an already available and therefore external universality, as an uninterrupted process of coordinating and setting in reciprocal relations. It is the latter process which is the true “generator” of structures as constantly under construction and reconstruction. The subject exists because, to put it very briefly, the being of structures consists in their coming to be, that is, their being “under construction.” [Ibid. p. 140]

Piaget’s structuralism must be considered as a positive force in the structural movement but, my “being,” when considered merely as a being in a continual state of “structuring structures,” seems to me at least, a somewhat sterile notion, especially a moments like bedtime when I tuck my children in for the night. As the lyrics of that gorgeous song by Peggy Lee suggests, “Is that all there is? Is that all there is my friend? Then lets…” If that is all there is, then perhaps we should hope Anthony Giddens was right when he declared structuralism to be a dead movement!

In addition to having an ahistorical and apolitical bent, structuralism also is criticized for its lack of aesthetic and moral values. Structuralism underestimates the significance of dialectics and social determinism. At times, structuralism brutally depersonalizes the individual and, in some cases, lacks the content that can be submitted to empirical confirmation. Levi-Strauss’s structural interpretation of myth comes to mind as an obvious example of an untestable theory. Having recognized structuralism’s flaws, however, there still must be something there to induce all this intellectual activity. I happen to believe there is. I would not have chosen structuralism as a paper topic for this class if I had thought otherwise. In the third part of this paper, I will describe what I believe this something is as I hope to address the above criticisms. But, before I get to the third part of the paper I want to briefly consider the structuralist (some might say post-structuralist) thought of Michel Foucault. Ironically (once agai
n), it is in the thought of a man who found little or no value in self that we might find the bridge to an unlimited resource of value and significance.

Godel’s Incompleteness Theorem-Reciprocal Movement

May 2, 2009

Between Syntax And Arithmetic-Godel’s Incompleteness Theorem-Reciprocal Movement (RM)

Foundational Crisis In Mathematics

Divergencies, convergencies and, at times, insurmountable paradox, has marked the history of mathematics. A major discontinuity in the history of mathematics occurred in the l880’s when Georg Cantor developed his theory of sets. Cantor developed a diagonal method from which he worked out the mathematics of infinities. The essence of this method is, according to Douglas Hofstadter, “the fact of using one integer in two different ways–or, one could say, using one integer on two different levels–thanks to which one can construct an item which is outside of some predetermined list.” [Douglas R. Hofstadter, Godel, Escher, Bach: an Eternal Golden Braid, 1979, p. 423]

The diagonal method developed by Cantor was not the problem, rather, the controversy sprang from what Cantor was able to prove with this method. For instance, in his investigations into non-denumerable sets Cantor was able to prove:

“Any line segment, no matter how small, contains as many points as an infinite straight line. Further, the segment, contains as many points as there are in an entire plane, or in the whole of space of n dimensions (where n is any integer greater than zero) or finally in a space of denumerable infinite number of dimensions.” [E.T. Bell, Men Of Mathematics, 1937, p571]

By bringing into question fundamental axioms such as: every magnitude is equal to itself; the part is less than the whole; two magnitudes, equal separately to a third, are equal to each other; etc., Cantor’s transfinite mathematics brought the very foundation of mathematics into question.

Three foundational schools of mathematics arose in response to Cantor’s transfinite mathematics. Each produced their own point of view concerning the nature of number. Bertrand Russell advocated that mathematics could be reduced to a system of logic while the school of thought representing intuitionalism accepted number as a “given”. Kronecker, a spokesperson for the intuitionalist school, is reported to have said, “The whole numbers were made by the good Lord, everything else is the work of man.” [A.S. Luchins and E.H. Luchins, Logical Foundations Of Mathematics For Behavioral Scientists, 1965, p.56] Not surprisingly the intuitionalists took exception to many of the concepts associated with the mathematics of infinitesimals.

In an attempt to end the controversy over number, the third foundational school, the formalists (Hilbert), set themselves the task of creating an axiomatic language (using methods acceptable to both logists and intuitionalists) that would define a formal number-theoretic system (a system inclusive of the natural numbers, analysis and calculus). If successful, this axiomatic language would have been both consistent and complete and it would have eliminated the paradoxes and antinomies that were found in set theory and classical mathematics. For better or worse, the formalist school of thought did not achieve their goal. What was achieved however was a 1931 paper by Kurt Godel which determined the goals of the formalist approach to mathematics to be unattainable. In his paper, entitled, On Formally Undecidable Propositions of TNT (typographical number theory) and Related Systems, Godel offered a proof which demonstrated that given any sufficiently powerful formal system, it is not possible for it to be complete and, while it may be consistent, its consistency is not provable within the system.

When The Foundation Of Mathematics Is Based On Indeterminate Numbers

Between Content And Form, Between Assimilation And Accommodation Are The Polar Phases Of The Total Life Process—And Reciprocal Movement (RM)

There are a number of important consequences that follow from Godel’s proof but the one that is most significant in respect to Piaget’s constructionist structuralism, it seems to me, is the one that implies the existence of supernatural numbers. The derivation of a true theorem in number theory, which is its own negation, may be interpreted as the requisite condition for the existence of supernatural numbers. These numbers are peculiar in that they have the property of being infinitely large while possessing no numeral representation. But, everything that can be proven for natural numbers can be proven for supernatural numbers, with the exception that natural numbers are determinate while supernatural numbers are indeterminate.

When the foundation of mathematics is considered in this light, Piaget becomes easier to comprehend. For instance, when Piaget describes the nature of knowledge as being like a pyramid of knowledge that “no longer rests on foundations but hangs by its vertex, an ideal point never reached and, more curious, constantly rising,” he is being very consistent with the latest developments in number theory. Piaget continues:

“In short, rather than envisaging human knowledge as a pyramid or building of some sort, we should think of it as a spiral the radius of whose turns increases as the spiral rises…This means, in effect, that the idea of structure as a system of transformations becomes continuous with that of construction as continual formation.” [Jean Piaget, Structuralism, 1970, p. 34]

The concept of “knowledge as structure” and structure as a “double movement” e.g., the “completeness proof” for natural numbers requiring supernatural numbers and the “consistency proof” for supernatural numbers requiring natural numbers, becomes generalized in the thought of Piaget as the relationship of interdependence that exists between content and form. It is in this interdependent relationship that we find the basis for Piaget’s constructionist theory. Piaget puts this conclusion in his own words:

“Since Godel,…the idea of a formal system of abstract structures is thereby transformed into that of the construction of a never completed whole, the limits of formalization constituting the grounds for incompleteness, or, as we put it earlier, incompleteness being a necessary consequence of the fact that there is no “terminal” or “absolute” form because any content is form relative to some inferior content and any form the content for some higher form.” [Piaget, Structuralism, 1970 p. 140]

With the knowledge of the interdependence of content and form firmly established, Piaget turns to the subject of Biology in order to give his theory a concrete elaboration.

Piaget was deeply influenced by Darwinian evolution. In Piaget’s understanding of processes and states, in terms of developmental stages, we see the depth of this influence. However, Piaget went further than Darwin in his elaboration on the effects of the environment on the development of the individual. In addition to natural selection, Piaget believed something more was going on in an organisms adaptation to its environment. For Piaget an active restructuring and accommodation of an organism to its environment also influenced the development of the organism. Piaget described this new level of accommodation in terms of the interplay of accommodation and assimilation that occurs in the normal life processes. Martindale explains:

“Basic to all Piaget’s explanations is a conception of the individual life process. This has two major aspects or phases: the assimilation of objects to individual activity, on the one hand, and the accommodation of activity to the object world on the other. The two processes of assimilation and accommodation are polar phases of the total life process. They are not always in equilibrium and may even operate in partial autonomy from one another. One of the fundamental facts of the life proces
s is the establishment of an equilibrium between assimilation and accommodation.” [Don Martindale, 1981, p. 342