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Final Report  : Section 1

ARL Contract

TelArt Inc., Chantilly Virginia

December 31, 2000

Full Paper


In May 2000, Zenkin and Prueitt discussed the possibility that there might be a common scheme in the use to CCG technology in new knowledge generation. This discussion occurred after Dr. Prueitt had developed an appreciation of the Computer Cognitive Graphics (CCG) technology as it is applied to number theory. Drs Art Murray, Alex Citkin, Peter Kugler, Bob Shaw, Michael Turvey, and Kevin Johnson have assisted him in this evaluation.  Our first project report to the ARL delineates this understanding.

Our final report to ARL generalizes the CCG technology. From the generalized CCG it becomes clear that the re-application of the original CCG technology to other objects of investigation can be achieved if and only if all real aspects of the syntactic and semantic representational problems are addressed.

The claim made in this ARL Report is that the CCG technology, as applied to number theory, has addressed all real aspects of the syntactic and semantic representational problems. A specific object of investigation, this being the theorems of classical number theory, shapes how these aspects are addressed. No semantic issues exist, except as noted in the problem of induction. Thus the mass of the CCG technology is merely formal and syntactic. The core, however, is essential and this core is about how one manages mental induction. 

We hold that induction is not and cannot be considered algorithmic in nature. We cite Western Scholarship Robert Rosen and Roger Penrose work as well as the work of J. J. Gibson and Karl Pribram.  Thus the interface between algorithm computers and human mental activity is necessary in any generation of new knowledge.  It appears that much of Russian Applied Semiotics is based on this Peircean concept that an interpretant is actually required during the generation of new knowledge.

Section 1: A review of the CCG application to number theory and its generalization

Specifically, elementary number theory is a formal construct that is built on the Peano axiom, the additive and multiplicative operators, and on the use of a principle of mathematical induction. The CCG technology can then be seen to have the following parts:

A representational schema

A visualization schema

An induction schema

In number theory, the representation allows color codes to represent division properties. In classical number theory, division does not induce non-integers, but rather truth evaluation of whether a remainder is zero, (or any other integer - depending on the way the theorem is stated). Now it is important to state that Dr. Prueitt has some background in number theory and as a consequence of seeing number theory in a new light, there appears to be new theorems regarding invariances of residues. Perhaps others can see these new theorems also. There is additional work that would have to be achieved in order to tease a the specific statement about these theorems.

The possibility of new mathematics is pointed out only because there is a mental effort required to think about how color representation fits into the CCG methodology. That this effort has lead to an intuition about new mathematics is a statement that must be taken on a plausibility argument. This plausibility is ultimately how the CCG methodology should be judged.  Although positive results of various types exist and can be shown, the conceptual grounding for CCG remains foreign, not only due to Russian origin but primarily due to the misrepresentations made by modern Artificial Intelligence regarding the nature of human induction. 

There is a suggestion that there are some new theorems that are delineated by Prueitt's mental intuition when the cognitive effort is made to describe the CCG techniques.  Prueitt is willing to discuss this issue at the proper time.  However, the existence of new theorems in number theory is not of immediate interest.  Prueitt claims that the theory of algebraic residues is not completely developed, and that pure mathematicians who know this field well will be able to quickly see the same intuition.  This intuition comes immediately from the realization of how color-coding is used in the CCG applied to number theory.

How would CCG assist in our experiencing intuitions?

An induction is to be established by the physical representation (by colors in this case) and the subsequent representation of truth/falseness evaluation of specific properties of a generated sequence of numbers using the 2 dimensional grid. Zenkin has many examples of how this has worked for him. Prueitt sees a different class of theorems because he has a different mathematical training and internal percepts about the Peano axiom and the additive and multiplicative operators. Any other pure mathematician would see theorems that are new, depending on the nature of the intuitions that are resident in the mind of the pure mathematician.  CCG would be useful in the completion of mathematical reasoning from whatever experience the mathematician might have.

Prueitt holds that any formal system can be vetted using slight modifications of the CCG representation and visualization demonstrated by Zenkin. If this claim is correct, then areas of abstract algebra would fall under the technique. The requirement is that a human has deep intuitions about an object of investigation and that the representation and visualization setup a route to induction regarding truth / falseness of theorems (see the work of Russian father of quasi axiomatic theory, Victor Finn, on routs to induction).

This means that someone who is deeply involved with algebra and who studied the CCG applications to number theory would likely begin to (immediately) see how to represent and visualize relationships such as the property of being a generator of a semi group. Once this new mental intuition is established, then a principle of induction is required that allows the validation, or falsification, of intuitions. In the past application of CCG to number theory, this validation of intuition is equivalent to a formal proof, and yet is made using a proxy that is visual in nature. The proxy is established via the notion of a super-induction where the visual observation of a property transfers to a formal declaration of fact. This transfer is the core of the CCG technology and is not dependent on the specific representation or the visualization, as long as the visualization schema matches (completely) all syntactic and semantic representational problems.

In formal systems, the problem of syntactic and semantic representation is not only simple; but is also complete. There is little or no semantic dimension. Only the truth evaluation is semantic and this semantic evaluation is incompletely represented in the iterated folding of syntactic structure (via rules of deductive inference). In essence, one can almost claim that the only semantic aspect about number theory is that someone who sees the elegance of it can experience it as beautiful. The caveat is captured by the Godel theories on completeness and consistency, and on related notions communicated by Cantor and others (including Robert Rosen’s work on category theory). Of course, Zenkin is one of those who have advanced a disproof of Cantor's argument regarding the categorical non-correspondence between the whole numbers and the real numbers.

Prueitt reads this disproof in a certain way. The argument that Canton's diagonalization theory is flawed is really a comment on the nature of common mathematical induction. As Kevin Johnson has pointed out, there are many many ways to perform an induction. The common mathematical induction simply depends on an ordering of theorems in such a way that the tail of this sequence of theorems has invariance with respect to the truth evaluation.  The CCG representation and visualization simply allows a pure mathematician a by-pass of all orderings except one that results in visualization of the targeted invariance of a tail of a sequence of theorems.  This by-pass is non-algorithmic and thus must be managed by a human.

One can see this as a search space problem. In many cases modern computer science has identified what are called NP-complete problems. The NP-complete problem can be proved not to be computably solved with the iterative application of the folding (application) of the fundamental axioms and properties in the set up of the formal system. However, visual acuity by a human might see a route to a solution. In fact, Prueitt has made the argument that biological systems have evolved in such a way as to by-pass NP complete problems. He claims that the capacity for seeing a solution that cannot be computed is fundamental to biological intelligence.

In formal systems, the by-pass is simply a lifting away from and a replacement into the formal construct. This there is still no semantic dimension to the solution. This concept of lifting is consistent with Brower’s notion of intuition (Bob Shaw – private communication).  This means that the solution, once found, to NP-complete problems can then be proved using common inference and common induction. It is just a question of skipping and reordering.

Possible application to EEG and stock market data analysis

Zenkin and Prueitt were hoping that EEG data could be easily found with expert opinions about differential meaning of data patterns in context. Due to the uncertainty of how we might precede Prueitt did not pursue a collaborative relationship with EEG experts in Karl Pribram's lab or in any other lab. Such collaboration requires that the method we have devised for visualization be well developed and that our collaborative project with the Russians be well funded.

As we worked on this issue, it became clear that we could describe such a method only if the communication between Russia and the United States was better.  We need to involve neuroscientists both in St. Petersburg (Juri Kropotov) and in the USA (Karl Pribram). 

Given our limited resources, Prueitt decided to attempt to generalize the CCG methodology and then project this generalization back onto some object of investigation. The idea was that the generalization and separation of parts of the CCG techniques would show us how to proceed.

What we needed to figure out first was how to characterize the CCG method in such a way that aspects of the method could be separated into functional parts. Then each part might be generalized and then projected into a new use case.

We were open to possible investment directed at using indices in the analysis of stock market performance. This possibility still exists. However, it is felt that this application is unwise and not directed at a scientific or mathematical objective.

However, our thinking about the markets allowed us to see, for the first time, that we needed to have an Image Library. We needed a repository for the consequences of the evocation of knowledge about, or an intuition about, the past or future performance of the market. At this point, the work of other Russian applied semioticians (Pospelov and Finn) come into play. The Library becomes a repository for a system of tokens, each token deriving token meaning from intuitions vetted by the CCG representation and visualization, and confirmed by an induction. The system is then a formalism that is open to human manipulation as well as formal computations. The formalism has both a first order and a second order (control or tensor) system.

We have come face to face with the core difference between a formal system, like number theory or algebra, and a natural object of investigation, like the stock market. It is this difference that is ignored by most Western mathematicians and computer scientists. It is also this difference that illuminates the nature of Russian applied semiotics. The case of this assertion will not be full made here; as to some extent the assertion is ultimately a statement of belief.

In any case Zenkin and Prueitt both agreed that an Image Library might be built as a type of Artifact Warehouse, where the artifacts were the consequences of a super-induction mediated by some representational and visualization schema.

The problems are then defined as

How does one represent the object of investigation

How does one visualize the accrual of invariance

How does one establish conditions of induction

Prueitt has some experience with scatter gather methods used in the standard methods for vectorization of text. Thus he chose a collection of 312 Aesop fables to be his target of investigation. This choice was a secondary choice, since TelArt Incorporated continued to hope that situations in Russian might allow Alex Zenkin the time required to make a paper on his own attempted application of CCG to scientific data of some sort. As the deadline for our Final Report neared, it became clear that Prueitt would have to write the Final Report without additional original work from Russia.

In the next sections, results of the generalization of CCG are applied to the problem of parsing text. Text parsing ultimately is to be applied to a routing of information or a retrieval of information. Prueitt is designing a system for a worldwide evaluating of Indexing Routing and Retrieval (IRR) technologies, and thus the use of Prueitt's background was capitalized on for the purpose of completing our contractual obligations.

The URL announcing this IRR evaluation is at: link