Ph. D. & Dr. Sc. Lev Gelimson's Overmathematics Essence

Overmathematics Essence

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

11 (2011), 25

UDC 501:510

2010 Math. Subj. Classification: primary 00A71; second. 03E10, 03E72, 08B99, 26E30, 28A75.

Keywords: Overmathematics, constructive philosophy, reasonable fuzziness, symbolic existence, tolerable simplicity, perfect sensitivity.

Introduction

Classical mathematics [1] with hardened systems of axioms, intentional search for contradictions and even their purposeful creation cannot (and does not want to) regard very many problems in science, engineering, and life. This generally holds when solving valuation, estimation, discrimination, control, and optimization problems as well as in particular by measuring very inhomogeneous objects and rapidly changeable processes. It is discovered [2] that classical fundamental mathematical theories, methods, and concepts [1] are insufficient for adequately solving and even considering many typical urgent problems. The very fundamentals of classical mathematics have evident shortcomings and lacks. There are gaps between the real numbers, and the probabilities of many reasonable possible events vanish or do not exist at all. In each concrete (mixed) physical magnitude, there is no known operation. Sets are no models for many objects collections without structure. Each measure has a very restricted domain of sensitivity. The absolute error alone is noninvariant and insufficient for quality estimation. The relative error is uncertain in principle and has a very restricted domain of applicability. The unique known method applicable in principle to contradictory (e.g., overdetermined) problems typical in any data processing is the least square method (LSM) [1] that has narrow applicability and adequacy domains as well as many fundamental defects. For estimating the quality of distribution approximations, there is no applicable proposition at all. Classical mathematics insufficiency leads to unreliable results, time and cost loss, dangers, accidents, and catastrophes.

Exclusively Constructive Creative Philosophy Principles as a System of Revolutions in Philosophy

Fundamental principles of exclusively constructive creative philosophy build a fundamental system of revolutions in philosophy, in particular, the following subsystems.

1. Fundamental Principles of Exclusively Constructive Creative Philosophy as a Fundamental Subsystem of Revolutions in Philosophy

The fundamental subsystem of revolutions in philosophy includes the following fundamental principles of exclusively constructive creative philosophy:

1. Exceptional natural constructivism (with the complete absence of artificial destructiveness).

2. Useful free creativity (exclusively practically purposeful, verified, and efficient unlimitedly free creativity, intuition, and phantasy flight).

3. Scientific optimism and duty (each urgent problem can and must be solved adequately and efficiently enough).

4. Complication utilization (creating, considering, and efficiently utilizing only necessary and useful also contradictory objects and models, as well as difficulties, problems, and other complications).

5. Symbolic feasibility (at least symbolic existence of all the necessary and useful even contradictory objects and models).

2. Advanced Principles of Exclusively Constructive Creative Philosophy as an Advanced Subsystem of Revolutions in Philosophy

The advanced subsystem of revolutions in philosophy includes the following advanced principles of exclusively constructive creative philosophy.

1. Exclusively useful intuitive evidence and provability (reasonable fuzziness, intuitive ideas without axiomatic rigor if necessary and useful).

2. Unrestrictedly flexible constructivism (if necessary even creating new knowledge (concepts, approaches, methods, theories, doctrines, and even sciences) to adequately set, consider, and solve urgent problems).

3. Tolerable simplicity (choosing the best in the not evidently unacceptable simplest).

4. Perfect sensitivity, or conservation laws universality (no uncompensated change in a general object conserves its universal measures).

5. Exact discrimination of noncoinciding objects and models (possibly infinitely or overinfinitely large with infinitesimal or overinfinitesimal distinctions and differences).

3. Some Other Principles of Exclusively Constructive Creative Philosophy

Among other principles of exclusively constructive creative philosophy are the following:

1. Truth priority (primacy of practically verified purely scientific truths and criteria prior to commonly accepted dogmas, views, agreements, and authority, with all due respect to them).

2. Peaceful pluralism (with peaceful development of scientific and life diversity).

3. Useful creative inheritance (efficiently using, analyzing, estimating, and developing already available knowledge and information).

4. Useful constructive freedom (unrestrictedly free exclusively constructive and useful self-determination and activity, in particular, in knowledge and information research, creation, and development).

5. Fundamentality priority (primacy of conceptual and methodological fundamentals).

6. Knowledge efficiency (only useful quality (acceptability, adequacy, depth, accuracy, etc.) and amount (volume, completeness, etc.) of knowledge, information, data, as well as creation, analysis, synthesis, verification, testing, structuring, systematization, hierarchization, generalization, universalization, modeling, measurement, evaluation, estimation, utilization, improvement, and development of objects, models, knowledge, information, and data along with intelligent management and self-management of activity).

7. Mutual definability and generalizability (relating successive generalization of concepts in definitions with optional linear sequence in knowledge construction).

8. Useful unification of opposites only conditionally distinguished (such as real/ideal, specific/abstract, exact/inexact, definitively/possibly, pure/applied, theory/experiment/practice, nature/life/science, for example, the generally inaccurate includes the inaccurate as the limiting particular case with the zero error).

9. Partial laws sufficiency (if there are no known more general laws).

10. Focus on discoveries and inventions (dualistic unity and harmony of academic quality and originality, discovering phenomena of essence, inventive climbing, helpful knowledge bridges, creative multilingualism, scientific art, anti-envy, learnability, teachability, and terminology development).

Principles of Unimathematics as a System of Revolutions in the Principles of Mathematics

The principles of exclusively constructive creative unimathematics (mega-overmathematics) constitute a system of scientific revolutions in the principles of mathematics including the following subsystems.

1. Fundamental Principles of Unimathematics as a Fundamental Subsystem of Revolutions in the Principles of Mathematics

The fundamental subsystem of revolutions in the principles of mathematics includes the following principles of unimathematics:

1. Typical urgent problems priority and exclusiveness (adequately setting and solving and efficiently using urgent problems only with completely avoiding unnecessary considerations is the only criterion of the necessity and usefulness of creating and developing new knowledge including concepts, approaches, methods, theories, doctrines, and sciences).

2. Intuitive conceptual and methodological fundamentality priority (creating and efficiently using unified knowledge foundation due to fundamental general systems including objects, models, and intuitive fuzzy principles, concepts, and methodology).

3. Reasonable fuzziness with useful rigor only (exclusively practically useful axiomatization, deductivity, and rigorously proving, as well as intuitive ideas without axiomatic strictness if necessary and useful).

4. Unrestrictedly flexible constructivism (even creating new sciences to adequately set, consider, and solve typical urgent problems).

2. Noncontradictoriness Principles of Unimathematics as a Noncontradictoriness Subsystem of Revolutions in the Principles of Mathematics

The noncontradictoriness subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. Unifying membership, inclusion, and part-whole relations.

2. Necessary and useful creation exclusiveness (efficiently and intelligently creating and considering exclusively necessary and useful objects and models with completely ignoring any artificial contradictions typical in classical mathematics).

3. Efficiently utilizing contradictoriness and other complications (creating, considering, and efficiently utilizing exclusively necessary and useful contradictory objects and models, as well as difficulties, problems, and other complications).

4. Symbolic feasibility (at least symbolic existence of all the necessary and useful even contradictory objects and models).

5. Decision-making delay (if necessary and useful, e.g. by estimating existence and sense with a possible further revaluation in the course of review).

3. Universalization Principles of Mega-Overmathematics as a Universalization Subsystem of Revolutions in the Principles of Mathematics

The universalization subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. Infinite cardinals canonization (infinite cardinal numbers as canonical positive infinities).

2. Zeroes reciprocals overinfinities canonization (signed zeroes reciprocals as canonical overinfinities).

3. Hyper-Archimedean axiom (a natural generalization of the Archimedes axiom to the infinite and the overinfinite).

4. Exactness of the infinite and the overinfinite (perfectly sensitive, invariant, and universal infinite and overinfinite, infinitesimal and overinfinitesimal generalization of the numbers by the uninumbers with exact measurement generalizing counting, unlimited (possibly even noninteger and uncountable) manipulation and operability, as well as exact discrimination in the infinite and the overinfinite even by infinitesimal and overinfinitesimal distinctions and differences).

5. General (nonlogical) quantification (assignment, definition, determination, and measurement of the individual quantity of a element becoming a quantielement and of the individual quantities of elements in a set which becomes a quantiset).

6. Perfect manipulation (perfectly sensitive, invariant, and universal useful modeling, expression, evaluation, counting measurement, estimation, and essential generalization of urgent objects, relations, structures, systems, and their contents extending sets and quantisets).

7. Conservation laws universality (in the overinfinitesimal, the infinitesimal, the finite, the infinite, and the overinfinite).

4. Efficiency Principles of Mega-Overmathematics as an Efficiency Subsystem of Revolutions in the Principles of Mathematics

The efficiency subsystem of revolutions in principles of mathematics includes the following principles of unimathematics:

1. Uniproblem unisolvability (existence and expressibility of the best quasisolution, solution, and supersolution among possibly inexact meaningful pseudosolutions to any urgent uniproblem with setting as a unisystem with unknown unisubsystems).

2. Tolerable simplicity (selecting the best in the class of not evidently unacceptable simplest meaningful pseudosolutions).

3. Efficient knowledge (efficient quality (acceptability, adequacy, profundity, exactness, structurality, systematization, inheritance, universality, invariance, strength, stability, reliability, flexibility, etc.) and quantity (volume, completeness, etc.) of objects, models, knowledge, information, data, and their perfectly sensitive creation, analysis, synthesis, verification, testing, structuring, systematization, hierarchization, generalization, universalization, modeling, evaluation, measurement, estimation, utilization, improvement, development, and reasonable control).

4. Free intuitive intelligent iterativity (possibly with many sources and directions, unrestrictedly flexible universal algorithms with avoiding computer zeroes and infinities and independent of analytic solvability with providing mapping contractivity).

5. General noncriticality (subcritical, critical, and supercritical states, processes, and phenomena in a general structured system which are defined and determined by generally noncritical relationships).

6. General nonlimitability (underlimiting, limiting, and overlimiting states, processes, and phenomena in a general structured system which are defined and determined by generally nonlimiting relationships).

Overmathematics

Overmathematics [2] created, advanced, and developed by the author is based on the constructive philosophy and its own principles of and includes dozens of principally new general theories and methods. Moreover, it discovers very many new possibilities, levels, and horizons fully unimaginable and unthinkable in classical mathematics to solve urgent general problems. Overmathematics provides discovering new phenomena and laws of nature, society, and consciousness. There are many urgent scientific and life problems that permit no rigorous approach and call for creating constructive philosophy and overmathematics as its universal language. Their fundamental principles and presentation must be fuzzy unlike any axioms system. Such a fuzziness and considering no optional relation in any problem eliminate any destruction in overmathematics that seems to be not only science but also art and even life.

The overmathematics system includes overmathematics itself as a universal language of new scientific thinking and other fundamental sciences (of quantielements, quantiobjects, quantioperations, quantirelations, quantistructures, quantisystems, quanticonditions, quantistates, quantiprocesses, and quantilaws) adequately expressing many urgent general objects in nature, society, and thinking. Unified theories of general dimensions, infinities, infinitesimals, quasinumbers, and supernumbers complete the universal number scale for perfectly expressing arbitrary uniquantities. Unified theories of general meaners, bounders, truncators, measurers, integrators, probabilers, errers, and reservers give subtle general estimators, discriminators, and controllers for inexact, approximate, and even exact general objects. Unified theories and general methods of reasonably simulating and effectively solving many urgent fundamental and applied general problems determine their best analytic quasisoltitions and even supersoltitions in their general pseudosoltitions. Overmathematics and other fundamental mathematical and strength sciences give a hierarchical quantisystem of the new general laws of the superuniverse.

The proposed basic concepts and solving methods are universal and very effective perhaps by solving any general problem in science, engineering, and life and especially urgent in the case of responsible objects under dynamically changeable extreme conditions.

References

[1] Encyclopaedia of Mathematics / Managing editor M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994.

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2004, 496 pp.