Ph. D. & Dr. Sc. Lev Gelimson: Fundamental Defects of Classical Applied Mathematics

Fundamental Defects of Classical Applied Mathematics

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Monograph

The “Collegium” All World Academy of Sciences Publishers

Munich (Germany)

2012

Keywords: Applied science, 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.

Introduction

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.

Mega-overmathematics [2] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further mega-overmathematics fundamental sciences systems developing, extending, and applying overmathematics. Among them are, in particular, science unimathematical test fundamental metasciences systems [3] which are universal.

Applied Science Unimathematical Test Fundamental Metasciences System

Applied science unimathematical test fundamental metasciences system in mega-overmathematics [2] is one of such systems and can efficiently, universally and adequately strategically unimathematically test any pure science. This system includes:

fundamental metascience of applied science test philosophy, strategy, and tactic including applied science test philosophy metatheory, applied science test strategy metatheory, and applied science test tactic metatheory;

fundamental metascience of applied science consideration including applied science fundamentals determination metatheory, applied science approaches determination metatheory, applied science methods determination metatheory, and applied science conclusions determination metatheory;

fundamental metascience of applied science analysis including applied subscience analysis metatheory, applied science fundamentals analysis metatheory, applied science approaches analysis metatheory, applied science methods analysis metatheory, and applied science conclusions analysis metatheory;

fundamental metascience of applied science synthesis including applied science fundamentals synthesis metatheory, applied science approaches synthesis metatheory, applied science methods synthesis metatheory, and applied science conclusions synthesis metatheory;

fundamental metascience of applied science objects, operations, relations, and criteria including applied science object metatheory, applied science operation metatheory, applied science relation metatheory, and applied science criterion metatheory;

fundamental metascience of applied science evaluation, measurement, and estimation including applied science evaluation metatheory, applied science measurement metatheory, and applied science estimation metatheory;

fundamental metascience of applied science expression, modeling, and processing including applied science expression metatheory, applied science modeling metatheory, and applied science processing metatheory;

fundamental metascience of applied science symmetry and invariance including applied science symmetry metatheory and applied science invariance metatheory;

fundamental metascience of applied science bounds and levels including applied science bound metatheory and applied science level metatheory;

fundamental metascience of applied science directed test systems including applied science test direction metatheory and applied science test step metatheory;

fundamental metascience of applied science tolerably simplest limiting, critical, and worst cases analysis and synthesis including applied science tolerably simplest limiting cases analysis and synthesis metatheories, applied science tolerably simplest critical cases analysis and synthesis metatheories, applied science tolerably simplest worst cases analysis and synthesis metatheories, and applied science tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;

fundamental metascience of applied science defects, mistakes, errors, reserves, reliability, and risk including applied science defect metatheory, applied science mistake metatheory, applied science error metatheory, applied science reserve metatheory, applied science reliability metatheory, and applied science risk metatheory;

fundamental metascience of applied science test result evaluation, measurement, estimation, and conclusion including applied science test result evaluation metatheory, applied science test result measurement metatheory, applied science test result estimation metatheory, and applied science test result conclusion metatheory;

fundamental metascience of applied science supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including applied science supplement metatheory, applied science improvement metatheory, applied science modernization metatheory, applied science variation metatheory, applied science modification metatheory, applied science correction metatheory, applied science transformation metatheory, applied science generalization metatheory, and applied science replacement metatheory.

The applied science unimathematical test fundamental metasciences system in megamathematics [2] is universal and very efficient.

In particular, apply the applied science unimathematical test fundamental metasciences system to classical applied mathematics [1].

Nota bene: Naturally, all the fundamental defects of classical pure mathematics [1] discovered due to the pure science unimathematical test fundamental metasciences system in megamathematics [2] also hold in classical applied mathematics [1].

Fundamental Defects of Classical Pure Mathematics

Even the very fundamentals of classical pure mathematics [1] have evident cardinal defects of principle.

1. The real numbers R evaluate no unbounded quantity and, because of gaps, not all bounded quantities. The same probability p_n = p of the random sampling of a certain n ∈ N = {0, 1, 2, ...} does not exist in R , since ∑_n∈N p_n 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.

7. 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.

Naturally, there are very many other lacks and shortcomings of classical pure mathematics [1]. For example, a power of a negative number is well-defined for even positive integer exponents only, see counterexamples

(-1)³ = -1 ≠ 1 = [(-1)⁶]^1/2 = (-1)^6/2 ,

(-1)^1/3 = -1 ≠ 1 = [(-1)²]^1/⁶ = (-1)^2/⁶ .

Therefore, the very fundamentals of classical pure mathematics [1] have a lot of obviously deep and even cardinal defects of principle.

Fundamental Defects of Classical Applied Mathematics

In the very fundamentals of classical applied mathematics [1] with its own evident cardinal defects of principle, 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.

8. 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.

9. 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.

10. 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. 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 best quasisolution illusion as the highest truth fully ungrounded and inadequate. Additionally consider the simplest least square method [1] approach which is typical. Minimizing the sum of the squared differences of the alone preselected coordinates (e.g., ordinates in a two-dimensional problem) of the graph of the desired approximation function and of everyone among the given data depends on this preselection, ignores the remaining coordinates, and provides no coordinate system rotation invariance and hence no objective sense of the result. Moreover, the method is correct by constant approximation or no data scatter only and gives systematic errors increasing together with data scatter and the deviation (namely declination) of an approximation from a constant. Therefore, the least square method [1] has many fundamental defects both in the essence (as causes) and in the applicability (as effects), is adequate only in some rare special cases and even in them needs thorough adequacy analysis. Experimental data are inexact, and their amount is always taken greater than that of the parameters in an approximating function often geometrically interpretable by a straight line or curve, plane or surface. That is why this method was possibly the most important one for any data processing and seemed to be irreplaceable.

11. 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.

Therefore, the very fundamentals of classical applied mathematics [1] have a lot of obviously deep and even cardinal defects of principle.

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. Science Unimathematical Test Fundamental Metasciences Systems. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 1.