Rule-Generating Key–The Gap Between Self And Self-Knowledge

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More MV Conversation

“Where did you get the idea for the X/Y form, anyway?” responded MV.

“I don’t know,” I said. “However, I was surprised to find Sartre’s philosophy in the middle of it. I never really liked Sartre. His philosophy was too abstract, too empty for my taste, but without it, I never would have discovered the X/Y form.”

“Wasn’t the ‘form’ about logic and levels?” said MV.

“Well, yes, but it’s even more about negation,” I responded, “It’s also about the negation that constitutes Sartre’s for-itself. From there, I guess, the idea more or less fell out of the self-referential paradoxes.”

“Self-referential paradoxes?”

“Yes, when definitions for particular entities relate back to the entity group to which the entity belongs,” I replied, “self-referential paradoxes result. The definitions that fall into the category of paradox are called self-referential concepts or relationships.”

“And these paradoxical relationships gave you the idea of the X/Y form?”

“Well, not exactly,” I replied. “But I guess they did make me think really hard about rules and what rules are based on, and even about the truth or certainty concerning what follows from rules. Just to give you some idea of what I’m talking about this is how Luchins, in his book on the foundations of mathematics, describes Russell’s antimony, plus a few other paradoxes:

[“By way of introduction to Russell’s antinomy, note that a set or collection of objects may or may not have the character of these objects. For example, a group of horses is not a horse and a group of people is not a person. But in some cases a set of objects does have the character of the objects. For example, a set of numbers may itself be a number. As another example, the set of all sets may be considered to be a set. In other words, a set may or may not be a number of itself. Thus, the set of all horses is not a member of itself where as the set of all sets is a member of itself. Consider now the set, denoted by S, of all sets that are not members of themselves. Is S a member of itself or not? Suppose S is a member of itself; then (by definition of S) S is not a member of itself. Suppose S is not a member of itself; then (by definition of S) it is a member of S. Hence we arrive at the contradiction that S is a member of S if and only if S is not a member of S. This is known a Russell’s antinomy and is an example of a set-theoretic antinomy…

In a similar illustration, consider the chief (or high priest) of a tribe who makes sacrifices for those and only those members of the tribe who do not make sacrifices for themselves. Does he make a sacrifice for himself? If he does, then he does not; and if he does not, then he does… (And lastly), another well-known antinomy concerns a liar. For example, suppose a person says, “This statement I am now making is a lie.” Is the quoted statement true or false? It can be shown that it can be neither, without involving a contradiction.” (Luchins and Luchins, Logical Foundations of Mathematics for Behavioral Scientists, p.13).]

“Your X/Y form fell out of this gobbley goop,” exclaimed MV.

“Let me back up a bit,” I replied. “I guess you could say that I am not the only one who ever held the belief that the universe is rational. Einstein believed in a rational universe, and so did the Greek philosopher Heraclites, who, some 2400 years ago, thought that a non-human intelligence or Logos ordered everything. Because the universe is rational, it is intelligible and an intelligible universe has a whole lot in common with the X/Y form, as it also does with self-reference too. To my way of thinking, the X/Y form is a pre-condition for self-reference to occur.

The original idea that there was a pre-condition for self-reference was developed in the symbolic logic of C.I. Lewis. To avoid contradictions such as occur in the Liar and other paradoxes, Lewis developed what he called pragmatic contradiction. It, pragmatic contradiction, treats together the speech and the act of speaking. “All statements are false” cannot be true because it implies, not a restriction against self-reference as Russell said, but because it implies the necessary truth of the contradictory opposite, “There exists at least one true statement.” Starting with a contradiction-free affirmation, the structures of knowledge can then be made to follow in a necessary and systematic fashion. In this way, the closed system problems that arise in mathematics are avoided. Trying to locate the foundation of this knowledge—the source of rule-generated information–was what led me to the idea of the X/Y form. What also led me to the X/Y form was that I saw how the “the pre-condition for self-reference,” the negation that constitutes Sartre’s for-itself, and the certainty of rule generated information, are all a consequence of the X/Y form.

My Response To—An Inherently Ra
tional Universe

Aug. 6, 1981

An Inherently Rational Universe

In my presentation on the different contexts of freedom, I was, basically, agreeing with Northrop’s realistic interpretation of how mathematical constructs are found to correspond with a real, knowable aesthetic universe. As Goethe emphasized and Northrop concurs, the cosmos is like a living organism even though law governs it. In other words, I was trying, in my presentation, to make sense out of the broad generalization that the universe is inherently rational. Support for this idea can be found in many areas:

In Jean Piaget’s exploration of cognitive development in children, and in Ernst Cassirer’s investigations—especially in his study of aphasia (Cassirer, 1957, p. 205-77), this idea does indeed find support. In fact, setting in front of me right now is a magazine article entitled, “The Wisdom Of Babies” (Newsweek, 1-12-81). It strongly suggests that an awareness of self is somehow embedded in the mind of babies. The article states: 1) Babies quickly develop the notion of ‘self’—of being different from other things in the world, which suggests that the brain may be prewired for this concept. 2) Infants are born with a set of dispositions, which may form the basis for symbolic thinking. 3) The belief that humans need language to make the mental leap from the specific to the general is becoming more and more questionable in that babies seem to be able to construct categories and make generalizations. Could it be that at the time of birth information is already being channeled by a set of rules? Perhaps, in the language of mathematics we can find what we are looking for—precise definitions for foundational rules.

Unfortunately, even in mathematics, foundational rules are hard to come by. Part of the problem is that what gets called mathematics in one culture is different from what gets called mathematics in other cultures. The source of the problem—how do you define number—has generated much controversy. In short, there is no agreement on whether number refers to an idea or to a property of objects or to a pencil stroke, and so on and so forth. Curiously, even with this hazy concept of number, mathematics still provides the language for the most exact of empirical disciplines—science. This problem has not gone unnoticed by mathematicians.

In the logic school of mathematics, number is defined in terms of the concept of sets and cardinal numbers. According to the Frege-Russell definition, “two sets are said to have the same cardinal number if there exists a one-to-one correspondence between them.” The cardinal number of a given set is defined as the set of all sets that have the same cardinal number as the given set. However, it was later shown that a contradiction arose from this number concept. Even before the contradiction arose, this definition was received poorly among intuitionists who did not consider it necessary to reduce the concept of natural number to simpler concepts. For them number was simply the result of the notion of an abstract entity plus the notion of an indefinite sequence of those entities.

An advocate of the formalist school of mathematics, David Hilbert, attempted to formalize mathematics in a way that would satisfy both the logicists and the intuitionists. Hilbert proposed to formulate classical mathematics as axiomatic theories and then prove that these theories were free from contradictions. This attempt came to an abrupt halt when Kurt Godel published a theorem that demonstrated that any formal number-theoretic system, if consistent, contains an undecidable formula; that is, a formula that can neither be proved nor disproved. In other words, Godel’s theorem tells us, “that it is self-contradictory to suppose that mathematics can be proved free from self-contradiction—that, in fact, there must always be true but unprovable theorems” (Pledge, 1959, p. 190). This striking result– Godel’s theorem, suggests that the source for the certainty of rule generated information will not be found in the “language of the universe,” which is what Galileo Galilee once called mathematics. If we can’t look to mathematics for certainty then where can we look?

Certainty is hard to find no matter where we look, but finding the origin of number shouldn’t pose so great an obstacle; at least that’s what the philosopher, mathematician, and linguist Ernst Cassirer thought. According to Cassirer, “…in many languages, the etymology of the first numerals suggests a link with the personal pronouns: in Indo-Germanic, for example, the words for ‘thou’ and ‘two’ seem to disclose a common root…we stand here at a common linguistic source of psychology, grammar and mathematics; that this dual root leads us back to the original dualism upon which rests the very possibility of speech and thought” (Cassirer, 1957, vol. 1, p. 244). The mathematician Dedekind traced the concept of number back to an even more fundamental origin. He ended up reducing the system of natural numbers to a single basic logical function: he considered the system to be grounded in “the ability of the mind to relate things to things, to make a thing correspond to a thing, or to image a thing in a thing (Cassirer, 1957, vol. 3, p. 257). If the origin of number is located in the mind’s ability to relate things to things, then the self-limiting theorems of mathematics, it seems to me, have something to say about consciousness itself, something strange, and, ultimately, something that will remain strange. Douglas Hofstadter seems to agree. In his book, Godel, Escher, Bach: An Eternal Golden Braid, he echoes this sentiment when he states:

“All the limitative Theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point that is the kiss of death: it guarantees that you can never represent yourself totally. Godel’s Incompleteness Theorem, Church’s Undecidability Theorem, Turing’s Halting Theorem, Tarski’s Truth Theorem—all have the flavor of some ancient fairy tale which warns you that “To seek self-knowledge is to embark on a journey which…will always be incomplete, cannot be charted on any map, will never halt, cannot be described” (1979, p. 697).

Perhaps, it is that crit
ical disjuncture between self and self-knowledge that holds the key to the rule-generating phenomenon that we are looking for!

Movement Here Does Not Occur Across Space And Time-It Occurs On the Back Of Negation

Response Continued


What if the self that we identify with, the one expressed in the word “I,” is something different from what we think it is, something that once it becomes identified will entail bold implications, not just for psychology, but also for everything that we know and understand.

In its simplest form, self-inquiry becomes a statement about X becoming aware of Y. The power of representation X by Y form, or, the self-awareness experience, didn’t materialize out of thin air; rather, it had to evolve like any other complex state of being. I am not talking about evolution in the usual sense here. I am talking about the evolution of self-consciousness, an evolution that evolves space and time (not the other way around) as products of its own evolution.

Obviously, a problem arises right away. How can consciousness evolve outside of space and time when change takes place inside space and time? In the same way that logical necessity exists independently of space-time, so too the existence of X/Y is not contingent upon the existence of space-time parameters. Logical necessity and logical possibility exist– inside and outside– space and time. Indeed, as the investigations of both Heidegger and Sartre have shown, the self—Dasein for Heidegger and the being of for-itself for Sartre—pre-exists temporality as the “condition for temporality.” That said it is still hard to imagine how change can take place outside of space and time. Perhaps Sartre can help here.

The negation-dependent being of Sartre’s for-itself provides insight into how the X/Y form can exist outside of time and space, and, in that circumstance, produce change. For Sartre, for-itself exists by virtue of its own negation as being-what- is-not–while-not–being-what-is. This peculiar way of existing is not so much different from the way X/Y exists outside of space and time. Just as, on the level of the for-itself, negation moves consciousness into an awareness of self (and temporality), so too, on the ground level of the X/Y form, the double negative moves X/Y into an affirmation of its own negative space.

Movement here does not occur across space-time; rather, it occurs, so to speak, on the back of negation. In a sufficiently complex condition, X/Y’s double negative form becomes affirmed in a higher state of being, albeit in the negative space of itself. In so far as this affirmation occurs in its own negative space—the negative space of a higher dimension of itself, this affirmation remains connected to the original X/Y form—X/Y’s pre-temporal, pre-spatial state of being. It is through this logical movement that X/Y will, eventually, gain space-time awareness, and, as Sartre has pointed out, become a question unto itself, but since all this rests on the X/Y form becoming more complex, I need to say a few words about the qualitative difference that increased complexity sometimes makes.

This idea has its critics, especially among those people who view the world in terms of strict causalities. Hofstadter, however, gives us a different view when he says, “Complexity often does introduce qualitative differences. Although it sounds implausible, it might turn out that above a certain level of complexity, a machine ceased to be predictable, even in principle, and started doing things on its own account, or to us a very telling phrase, it might begin to have a mind of its own” (1979, p. 389). But, we are not talking about machines here; we are talking about the changing flux of increasing complexity as it relates to logical form and space-time relationships. A complexity that, on one level, becomes an awareness of space—the space of an environment, and, on another level, becomes the awareness of the awareness of an event. In other words, both self and environment evolve out of the X/Y form.

From an evolutionary point of view, the X/Y form works like this: Beliefs are affirmed or rejected based on supporting reasons and evidence. Darwin’s theory of evolution, for instance, is accepted or rejected based on how evidence for Darwin’s theory is viewed. The process of accepting or rejecting reasoned evidence is what the X/Y form represents, but it can’t represent beliefs and evidence until it has evolved into the liberated state of (X/Y)-(X/Y)–(X/Y)(X/Y).

Freedom moves, but is limited by form. Evolution moves, but is limited by genetic mutations that may or may not benefit individuals or species. For the moment, let’s assume that understanding evolution requires not just understanding natural selection, but also understanding the liberation that occurs in the X/Y form. Unless inhibited, freedom moves in the direction of more freedom. But, the X/Y form tells us that freedom and form are bound together in the same way that structure and function are bound together, e.g. the heart muscle. In the original state of the X/Y form very little freedom is possible. However, in the evolution of liberation (the evolution of the X/Y form), when the X/Y form evolves it evolves as a whole unit.

When the power of representation X by Y form achieves the capacity to experience its own negative space, the X/Y form evolves–as a whole unit. By its other name, this liberated experience is called, “the emergence of life in a nurturing environment.” On the level of the X/Y form, life is manifested as “continuity occurring in discontinuity,” or, -(X/Y)(X/Y). This newly liberated form of duality, once again, upon achieving a sufficient level of complexity (now alive), experiences its own negative space. This negative space, by its other name, is called the embodied space of a physical event, the embodiment of (X/Y)-(X/Y), which, in turn, manifests “discontinuity occurring in continuity,” or, –(X/Y)(X/Y). Jean Paul Sartre described the experience of –(X/Y)(X/Y) as “for-itself consciousness,” but, when th
is experience is understood as the liberated state of (X/Y)-(X/Y)–(X/Y)(X/Y),—- what leaps fully formed from this new dimension of freedom is the capacity to accept or reject reasoned evidence for unsubstantiated assertions—the capacity for rational analysis. This idea will become clearer (I hope) as I continue to describe the logic behind how the X/Y form piggybacks itself into freer states of expressive awareness on the back of negation.

On its most fundamental level, the X/Y form expresses a double negation. On this level X and Y are entwined. In this process then, X/Y becomes –(X/Y)(X/Y) where –(X/Y) represents the negative space of a now affirmed (X/Y). This affirmed state, biologically speaking, lives in an environment—its negative condition. In death, this affirmation dissolves back into the entwined space of the X/Y form. But, in life, it maintains and perpetuates itself as it evolves more life– more freedom.

Once again, upon achieving a sufficient complexity, the affirmed state of X/Y liberates itself from its negative condition. At this higher level of affirmation, negation occurs in affirmation while occurring in affirmation occurring in negation—the negative space of itself. The X/Y form, in this second transmutation of itself, now becomes (X/Y)-(X/Y)–(X/Y)(X/Y) where (X/Y)-(X/Y) represents the negative space of the now affirmed state of (X/Y)(X/Y). This affirmed state, psychologically speaking, “knows itself” to be living in the space of a physical event—its negative condition. Here, the affirmed state of -(X/Y)(X/Y) becomes the person that references everything that lies outside of its own affirmation, which includes both the inner states of mind and the outer conditions that extend from the body– to the limits of the space-time continuum. In death, the person dissolves back into the whole of the dialectic–the X/Y form’s ground condition. But, on the rational level of “knowing events,” it maintains and perpetuates itself as more freedom evolves and more knowledge accumulates as a consequence.

I suppose, there is nothing to prevent this process from repeating, but, in real experience, I have no idea what that would mean. I do know, however, what (X/Y)-(X/Y)–(X/Y)(X/Y) means. The X/Y form has bootstrapped its own self-awareness, its own “I” awareness, and as such, the capacity of the self-representation X by Y form represents the uniquely human capacity to create and represent its own negative space. Human history, or what gets called civilization, is the direct result of (X/Y) liberating itself from–(X/Y), in the liberated form of (X/Y)-(X/Y)–(X/Y)(X/Y), be it in the form of the individual or the collective. And so the story of freedom and civilization continues, now in the form of culture, knowledge, and space/time, — the negative event of self-awareness.


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