Science Unimathematical Test Fundamental Metasciences Systems
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
Mechanical and Physical Journal
of the "Collegium" All World Academy of Sciences
Munich (Germany)
11 (2011), 7
Keywords: Megascience, revolution, megamathematics, overmathematics, unimathematical test fundamental metasciences system, knowledge, philosophy, strategy, tactic, analysis, synthesis, object, operation, relation, criterion, conclusion, evaluation, measurement, estimation, expression, modeling, processing, symmetry, invariance, bound, level, worst case, defect, mistake, error, reserve, reliability, risk, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, replacement.
There are many separate scientific achievements of mankind but they often bring rather unsolvable problems than really improving himan life quality. One of the reasons is that the general level of earth science is clearly insufficient to adequately solve and even consider many urgent himan problems. To provide creating and developing applicable and, moreover, adequate methods, theories, and sciences, we need their testing via universal if possible, at least applicable and, moreover, adequate test metamethods, metatheories, and metasciences whose general level has to be high enough. Mathematics as universal quantitative scientific language naturally has to play here a key role.
But 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.
Even the very fundamentals of classical mathematics [1] have evident lacks and shortcomings.
1. The real numbers R evaluate no unbounded quantity and, because of gaps, not all bounded quantities. The same probability pn = p of the random sampling of a certain n ∈ N = {0, 1, 2, ...} does not exist in R , since ∑n∈N pn is either 0 for p = 0 or +∞ for p > 0. It is urgent to exactly express (in some suitable extension of R) all infinite and infinitesimal quantities, e.g., such a p for any countable or uncountable set, as well as distributions and distribution functions on any sets of infinite measures.
2. The Cantor sets [1] with either unit or zero quantities of their possible elements may contain any object as an element either once or not at all with ignoring its true quantity. The same holds for the Cantor set relations and operations with absorption. That is why those set operations are only restrictedly invertible. In the Cantor sets, the simplest equations X ∪ A = B and X ∩ A = B in X are solvable by A ⊆ B and A ⊇ B only, respectively [uniquely by A = ∅ (the empty set) and A = B = U (a universal set), respectively]. The equations X ∪ A = B and X = B \ A in the Cantor sets are equivalent by A = ∅ only. In a fuzzy set, the membership function of each element may also lie strictly between these ultimate values 1 and 0 in the case of uncertainty only. Element repetitions are taken into account in multisets with any cardinal numbers as multiplicities and in ordered sets (tuples, sequences, vectors, permutations, arrangements, etc.) [1]. They and unordered combinations with repetitions cannot express many typical objects collections (without structure), e.g., that of half an apple and a quarter pear. For any concrete (mixed) physical magnitudes (quantities with measurement units), e.g., "5 L (liter) fuel", there is no suitable mathematical model and no known operation, say between "5 L" and "fuel" (not: "5 L" × "fuel" or "fuel" × "5 L"). Note that multiplication is the evident operation between the number "5" and the measurement unit "L". The Cantor set relations and operations only restrictedly reversible and allowing absorption contradict the conservation law of nature because of ignoring element quantities and hinder constructing any universal degrees of quantity.
3. The cardinality is sensitive to finite unions of disjoint finite sets only but not sufficiently sensitive to infinite sets and even to intersecting finite sets (because of absorption). It gives the same continuum cardinality C for clearly very distinct point sets in a Cartesian coordinate system between two parallel lines or planes differently distant from one another.
4. The measures are finitely sensitive within a certain dimensionality, give either 0 or +∞ for distinct point sets between two parallel lines or planes differently distant from one another, and cannot discriminate the empty set ∅ and null sets, namely zero-measure sets [1].
5. The probabilities cannot discriminate impossible and some differently possible events.
6. The operations are considered to be at most countable.
Further all existing objects and systems in nature, society, and thinking have complications, e.g., contradictoriness, and hence exist without adequate models in classical mathematics [1]. It intentionally avoids, ignores, and cannot (and possibly hence does not want to) adequately consider, model, express, measure, evaluate, and estimate many complications. Among them are contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, errors, information incompleteness, multivariant approach, etc. There were well-known attempts to consider some separate objects and systems with chosen complications, e.g., approximation and finite overdetermined sets of equations. To anyway consider them, classical mathematics only has very limited, nonuniversal, and inadequate concepts and methods such as the absolute error, the relative error, and the least square method (LSM) [1] by Legendre and Gauss ("the king of mathematics") with producing own errors and even dozens of principal mistakes. The same holds for classical mathematics estimators and methods.
The absolute error Δ [1] alone is noninvariant and insufficient for quality estimation giving, for example, the same result 1 for acceptable formal (correct or not) equality 1000 =? 999 and for inadmissible formal equality 1 =? 0. Further the absolute error is not invariant by equivalent transformations of a problem because, for instance, when multiplying a formal equality by a nonzero number, the absolute error is multiplied by the norm (modulus, absolute value) of that number.
The relative error δ [1] should play a supplement role. But even in the case of the simplest formal equality a =? b with two numbers, there are at once two propositions, to use either δ1 = |a - b|/|a| or δ2 = |a - b|/|b| as an estimating fraction. It is a generally inadmissible uncertainty that could be acceptable only if the ratio a/b is close to 1. Further the relative error is so intended that it should always belong to segment [0, 1]. But for 1 =? 0 by choosing 0 as the denominator, the result is +∞ , for 1 =? -1 by each denominator choice, the result is 2. Hence the relative error has a restricted range of applicability amounting to the equalities of two elements whose ratio is close to 1. By more complicated equalities with at least three elements, e.g., by 100 - 99 =? 0 or 1 - 2 + 3 - 4 =? -1, the choice of a denominator seems to be vague at all. This is why the relative error is uncertain in principle, has a very restricted domain of applicability, and is practically used in the simplest case only and very seldom for variables and functions.
The least square method [1] can give adequate results in very special cases only. Its deep analysis [2] by the principles of constructive philosophy, overmathematics, and other fundamental mathematical sciences has discovered many fundamental defects both in the essence (as causes) and in the applicability (as effects) of this method that is adequate in some rare special cases only and even in them needs thorough adequacy analysis. The method is based on the absolute error alone not invariant by equivalent transformations of a problem and ignores the possibly noncoinciding physical dimensions (units) of relations in a problem. The method does not correlate the deviations of the objects approximations from the approximated objects with these objects themselves, simply mixes those deviations without their adequately weighing, and considers equal changes of the squares of those deviations with relatively less and greater moduli (absolute values) as equivalent ones. The method foresees no iterating, is based on a fixed algorithm accepting no a priori flexibility, and provides no own a posteriori adapting. The method uses no invariant estimation of approximation, considers no different approximations, foresees no comparing different approximations, and considers no choosing the best approximation among different ones. And the second power itself cannot provide adequate results at all in almost any nonrivial case. These defects in the method essence lead to many fundamental shortcomings in its applicability. Among them are applicability sense loss by a set of equations with different physical dimensions (units), no objective sense of the result noninvariant by equivalent transformations of a problem, restricting the class of acceptable equivalent transformations of a problem, no essentially unique correction of applicability sense loss, possibly ignoring subproblems of a problem, paradoxical approximation, no analyzing the deviations of the result, no adequate estimating and evaluating its quality, no refining the results, no choice, and the highest truth ungrounded.
Further in classical mathematics [1], there is no sufficiently general concept of a quantitative mathematical problem. The concept of a finite or countable set of equations ignores their quantities like any Cantor set [1]. They are very important by contradictory (e.g., overdetermined) problems without precise solutions. Besides that, without equations quantities, by subjoining an equation coinciding with one of the already given equations of such a set, this subjoined equation is simply ignored whereas any (even infinitely small) changing this subjoined equation alone at once makes this subjoining essential and changes the given set of equations. Therefore, the concept of a finite or countable set of equations is ill-defined [1]. Uncountable sets of equations (also with completely ignoring their quantities) are not considered in classical mathematics [1] at all.
Megamathematics including overmathematics [2] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides efficiently, universally and adequately strategically unimathematically modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further megamathematics fundamental sciences systems developing, extending, and applying overmathematics. Among them are, in particular, science unimathematical test fundamental metasciences systems [3-17] including:
Science Unimathematical Test General Fundamental Metasciences System
Among science unimathematical test general fundamental metasciences are:
fundamental metascience of knowledge test philosophy, strategy, and tactic including knowledge test philosophy metatheory, knowledge test strategy metatheory, and knowledge test tactic metatheory;
fundamental metascience of knowledge consideration including knowledge fundamentals determination metatheory, knowledge approaches determination metatheory, knowledge methods determination metatheory, and knowledge conclusions determination metatheory;
fundamental metascience of knowledge analysis including subknowledge analysis metatheory, knowledge fundamentals analysis metatheory, knowledge approaches analysis metatheory, knowledge methods analysis metatheory, and knowledge conclusions analysis metatheory;
fundamental metascience of knowledge synthesis including knowledge fundamentals synthesis metatheory, knowledge approaches synthesis metatheory, knowledge methods synthesis metatheory, and knowledge conclusions synthesis metatheory;
fundamental metascience of knowledge objects, operations, relations, and criteria including knowledge object metatheory, knowledge operation metatheory, knowledge relation metatheory, and knowledge criterion metatheory;
fundamental metascience of knowledge evaluation, measurement, and estimation including knowledge evaluation metatheory, knowledge measurement metatheory, and knowledge estimation metatheory;
fundamental metascience of knowledge expression, modeling, and processing including knowledge expression metatheory, knowledge modeling metatheory, and knowledge processing metatheory;
fundamental metascience of knowledge symmetry and invariance including knowledge symmetry metatheory and knowledge invariance metatheory;
fundamental metascience of knowledge bounds and levels including knowledge bound metatheory and knowledge level metatheory;
fundamental metascience of knowledge directed test systems including knowledge test direction metatheory and knowledge test step metatheory;
fundamental metascience of knowledge tolerably simplest limiting, critical, and worst cases analysis and synthesis including knowledge tolerably simplest limiting cases analysis and synthesis metatheories, knowledge tolerably simplest critical cases analysis and synthesis metatheories, knowledge tolerably simplest worst cases analysis and synthesis metatheories, and knowledge tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;
fundamental metascience of knowledge defects, mistakes, errors, reserves, reliability, and risk including knowledge defect metatheory, knowledge mistake metatheory, knowledge error metatheory, knowledge reserve metatheory, knowledge reliability metatheory, and knowledge risk metatheory;
fundamental metascience of knowledge test result evaluation, measurement, estimation, and conclusion including knowledge test result evaluation metatheory, knowledge test result measurement metatheory, knowledge test result estimation metatheory, and knowledge test result conclusion metatheory;
fundamental metascience of knowledge supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including knowledge supplement metatheory, knowledge improvement metatheory, knowledge modernization metatheory, knowledge variation metatheory, knowledge modification metatheory, knowledge correction metatheory, knowledge transformation metatheory, knowledge generalization metatheory, and knowledge replacement metatheory;
fundamental metametascience of unimathematical test fundamental metasciences systems application including megascience development metatheory, megamathematics and megastrength fundamental mathematical, mechanical, strength, and physical sciences systems development metatheories.
Science Unimathematical Test Special Fundamental Metasciences System
Among science unimathematical test special fundamental metasciences are, e.g.:
Pure Mathematics Unimathematical Test Fundamental Metascience
Pure mathematics unimathematical test fundamental metascience includes:
metatheory of pure mathematics test philosophy, strategy, and tactic;
metatheory of pure mathematics consideration;
metatheory of pure mathematics analysis;
metatheory of pure mathematics synthesis;
metatheory of pure mathematics objects, operations, relations, and criteria;
metatheory of pure mathematics evaluation, measurement, and estimation;
metatheory of pure mathematics expression, modeling, and processing;
metatheory of pure mathematics symmetry and invariance;
metatheory of pure mathematics bounds and levels;
metatheory of pure mathematics directed test systems;
metatheory of pure mathematics tolerably simplest limiting, critical, and worst cases analysis and synthesis;
metatheory of pure mathematics defects, mistakes, errors, reserves, reliability, and risk;
metatheory of pure mathematics test result evaluation, measurement, estimation, and conclusion;
metatheory of pure mathematics fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.
Applied Mathematics Unimathematical Test Fundamental Metascience
Applied mathematics unimathematical test fundamental metascience includes:
metatheory of applied mathematics test philosophy, strategy, and tactic;
metatheory of applied mathematics consideration;
metatheory of applied mathematics analysis;
metatheory of applied mathematics synthesis;
metatheory of applied mathematics objects, operations, relations, and criteria;
metatheory of applied mathematics evaluation, measurement, and estimation;
metatheory of applied mathematics expression, modeling, and processing;
metatheory of applied mathematics symmetry and invariance;
metatheory of applied mathematics bounds and levels;
metatheory of applied mathematics directed test systems;
metatheory of applied mathematics tolerably simplest limiting, critical, and worst cases analysis and synthesis;
metatheory of applied mathematics defects, mistakes, errors, reserves, reliability, and risk;
metatheory of applied mathematics test result evaluation, measurement, estimation, and conclusion;
metatheory of applied mathematics fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.
Computational Science Unimathematical Test Fundamental Metascience
Computational science unimathematical test fundamental metascience includes:
metatheory of computational science test philosophy, strategy, and tactic;
metatheory of computational science consideration;
metatheory of computational science analysis;
metatheory of computational science synthesis;
metatheory of computational science objects, operations, relations, and criteria;
metatheory of computational science evaluation, measurement, and estimation;
metatheory of computational science expression, modeling, and processing;
metatheory of computational science symmetry and invariance;
metatheory of computational science bounds and levels;
metatheory of computational science directed test systems;
metatheory of computational science tolerably simplest limiting, critical, and worst cases analysis and synthesis;
metatheory of computational science defects, mistakes, errors, reserves, reliability, and risk;
metatheory of computational science test result evaluation, measurement, estimation, and conclusion;
metatheory of computational science fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.
Material Strength Science Unimathematical Test Fundamental Metascience
Material strength science unimathematical test fundamental metascience includes:
metatheory of material strength science test philosophy, strategy, and tactic;
metatheory of material strength science consideration;
metatheory of material strength science analysis;
metatheory of material strength science synthesis;
metatheory of material strength science objects, operations, relations, and criteria;
metatheory of material strength science evaluation, measurement, and estimation;
metatheory of material strength science expression, modeling, and processing;
metatheory of material strength science symmetry and invariance;
metatheory of material strength science bounds and levels;
metatheory of material strength science directed test systems;
metatheory of material strength science tolerably simplest limiting, critical, and worst cases analysis and synthesis;
metatheory of material strength science defects, mistakes, errors, reserves, reliability, and risk;
metatheory of material strength science test result evaluation, measurement, estimation, and conclusion;
metatheory of material strength science fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.
Object Strength Science Unimathematical Test Fundamental Metascience
Object strength science unimathematical test fundamental metascience includes:
metatheory of object strength science test philosophy, strategy, and tactic;
metatheory of object strength science consideration;
metatheory of object strength science analysis;
metatheory of object strength science synthesis;
metatheory of object strength science objects, operations, relations, and criteria;
metatheory of object strength science evaluation, measurement, and estimation;
metatheory of object strength science expression, modeling, and processing;
metatheory of object strength science symmetry and invariance;
metatheory of object strength science bounds and levels;
metatheory of object strength science directed test systems;
metatheory of object strength science tolerably simplest limiting, critical, and worst cases analysis and synthesis;
metatheory of object strength science defects, mistakes, errors, reserves, reliability, and risk;
metatheory of object strength science test result evaluation, measurement, estimation, and conclusion;
metatheory of object strength science fundamentals development, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement.
Science unimathematical test fundamental metasciences systems are universal and very efficient.
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.
[3] Lev Gelimson. Pure Science Unimathematical Test Fundamental Metasciences System. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 1
[4] Lev Gelimson. Fundamental Defects of Pure Mathematics. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 2
[5] Lev Gelimson. Applied Science Unimathematical Test Fundamental Metasciences System. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 3
[6] Lev Gelimson. Fundamental Defects of Applied Mathematics. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2012
[7] Lev Gelimson. Computational Science Unimathematical Test Fundamental Metasciences System. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 4
[8] Lev Gelimson. Fundamental Defects of Computational Sciences. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2012
[9] Lev Gelimson. Material Strength Science Unimathematical Test Fundamental Metasciences System. Physical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 1
[10] Lev Gelimson. Fundamental Defects of Material Strength Sciences. Physical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 2
[11] Lev Gelimson. Object Strength Science Unimathematical Test Fundamental Metasciences System. Physical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 3
[12] Lev Gelimson. Fundamental Defects of Object Strength Sciences. Physical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 4
[13] Lev Gelimson. Pure Megamathematics as a Revolution in Pure Mathematics. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2012
[14] Lev Gelimson. Applied Megamathematics as a Revolution in Applied Mathematics. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2012
[15] Lev Gelimson. Computational Fundamental Megascience as a Revolution in Computational Sciences. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2012
[16] Lev Gelimson. Material Megastrength Fundamental Sciences Systems as a Revolution in Material Strength Sciences. Physical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2012
[17] Lev Gelimson. Revolution in Object Strength Sciences. Physical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2012