Theoretical Research on Intelligence
Superfluid Math

The main focus of natural sciences nowadays is to locate and find out the function of nature’s intelligence behind all-natural phenomena as natural laws. For this reason in the cutting edge theories of physical sciences scientists are trying to formulate information and logical or intelligence based models and theories. But in their great hurry they neglected the fact that such a mathematical structure has already been discovered under the name: matrix logic.

The mathematical-physicist August Stern originally developed his higher dimensional logical theory to shed light on the intelligent or information-logical processes behind nature’s functioning, which he later on wanted to use for constructing intelligent physical systems and quantum computers. With further theoretical researches we will try to extend the power of matrix logic to prove that the precise formulations of intelligent processes and also consciousness will also mean full understanding of the nature of human intelligence and consciousness. This will also help us to understand logically and mathematically those higher intelligent functions which form the backbone of advanced idea formation. With this approach our Institute will try to give direct targets for today’s artificial intelligence initiatives and to keep these advanced researches human centered.

The Hungarian mathematician, József Kaczvinszky prepared the following short summary about the useful results and novelties of his so-called intrazerial mathematics (which could be translated in English as mathematics inside zero). His whole concept is based on his number-sphere theory which he developed as a direct consequence of his intrazerial analysis. This new number theory gives an interesting research avenue to see its effect on the foundation of mathematics, especially on algebraic number theory, representation theory and to prove the source of effectiveness behind the Hardy-Littlewood circle method as well as Ramanuja’s chain fraction methods and its universal extension in numerical analysis. The number-sphere theory also highlights more clearly the basis and function of Abraham Robinson’s non-standard analysis, and completes his idea of extending the real number line with infinitesimals and their reciprocals. These two together could help us for rediscovering the foundations of Vedic mathematics! This summary will help the professional mathematicians and theoretical scientist to get a glimpse or flavour of his genuine discovery.

“The number-spherical representation of sequences will never dissolve into a vague and uncertain infinity since even the number of the sequence members as well as the last term of the sequence could exactly be expressed on the sphere. In this way, even the summations of divergent sequences could be achieved. This result was the sole reason for introducing the new number-sphere concept while its other applied values could be listed as follows:

1. By giving a direct numerical expression for ∞ and 0 this new analytical system is able to clearly distinguish the infinitesimally small numbers from each other and also their dual or reciprocal ones. In classical mathematics ∞ and 0 are handled usually as symbols and can’t be located in any number systems. By our number-spherical theory however this could be achieved and by accepting the universal applicability of algebraic operations our intrazerial analysis could be formulated. We should note however, that the intrazerial system could not defy nowadays classical mathematics and does not change it in any way. It offers however high precision with adjusted order. Since classical mathematics’ zero order precision is just one of the limits of the total system, consequently, the classical system is just a limit case of the intrazerial one.

2. Instead of using the f(a)+f(a+1)+f(a+2)+… = S classical summation expression, which could be interpreted in many ways, the intrazerial analysis, by utilizing the number-sphere theory, expresses the other formula either as f(a)+f(a+1)+…+f(∞) sequence with infinite members or as a really unboundedly long sequence as f(a)+f(a+1)+…+f(I), where I is the symbol of unbounded Infinitum, which stays on the same number-sphere while running unboundedly. So, by using intrazerial analysis we are able to really distinguish the infinite member sequences from those which are extending unboundedly on the same number-sphere.

3. The necessity of convergence for counting limits or summations are completely eliminated from intrazerial analysis.

4. In case a summation formula for sequences with countable members is at hand, then by deciding which of the two expressions – mentioned above – should be used, its summation could be achieved even if the sequence is divergent, like 2+22+23+… ad inf = A.

5. It highlights the contradiction regarding geometric sequences expressed as f(a)+f(a+1)+f(a+2)+… = S, and then eliminate this contradiction.

6. It proves the non-existence of sequences of conditional limits and shows that their appearance in classical mathematics is due to a deceptive conclusion. In this way it extends the commutative law of additivity to conditionally limited sequences, like 1-½+⅓-…etc.

7. It gives a direct way for transforming any kind of definite integral formula to exact numerical summation expressions on the 0 unit number-sphere.

8. At the same time, it gives a technic to build up and calculate such integral formulas which could be solved through analogies in classical mathematics.

9. It defines the 0! = 1 equality through the number-spherical extension of the 1∙2∙3∙…∙2I sequence to unbounded Infinitum.

10. It gives sharp distinction between the summation of sequences in positive and negative direction.

11. It gives a clear vision on the real structure of infinite Fourier-sequences and it is able to refine them for high or infinitesimal precision.” (This could be applied to the so-called Langland-program, algebraic geometry and to define our unifying holomatrix concept).

12. In algebraic geometry it gives a unified method to express the different conical sections.

13. Discovers a new effect, the so-called number-distortion effect and defines its source and appearance. (This could be a linked to renormalisation effects in quantum-field theories).”

Group members:

István Dienes Principal Scientist (IARIP)
BSc., MSCI, control-engineer, matrix logic based quantum information modelling of intelligent conscious processes and self-awareness

Professor Diego L. Rapoport
PhD., mathematical-physicist, matrix logic based modelling of intelligence

Publications:
1. Diego L. Rapoport, June 06, 2010

Zoltán Szlávik
PhD. applied mathematician, machine sight and cognitive process modelling Near term
project plans