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

Fundamental Defects of Classical Advanced Mathematics

© 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)

12 (2012), 4

Keywords: Advanced mathematics, 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 [2-10] that classical fundamental mathematical theories, methods, and concepts [1] are insufficient for adequately solving and even considering many typical urgent problems.

Mega-overmathematics [2-10] 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 [11] which are universal.

Advanced Science Unimathematical Test Fundamental Metasciences System

The advanced science unimathematical test fundamental metasciences system [11] in mega-overmathematics [2-10] is one of such systems and can efficiently, universally and adequately strategically unimathematically test any advanced science. This system includes:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The advanced science unimathematical test fundamental metasciences system is universal and very efficient.

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

Fundamental Defects of Classical Advanced Mathematics

In particular, apply the advanced science unimathematical test fundamental metasciences system [11] to classical advanced mathematics [1].

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

1. Real Numbers

Abstract. 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.

Classical mathematics [1] believes there are no gaps between the real numbers whose set is denoted by R. The probabilities of many typical reasonable events do not exist, for example, the same probability

p_n = p

of the choice of a certain natural number n (e.g., 7) in the set

N = {0, 1, 2, ...}.

This might be proved as follows. If that probability were positive, the infinite sum of those probabilities for all natural numbers would be +∞. If that probability were zero then the countable sum of those probabilities for all natural numbers would vanish because each partial sum would be zero and the limit of the sequence of these sums would vanish. But this infinite countable sum is the probability of the reliable event, that some natural number is chosen, and must be exactly 1.

Further the probabilities of many typical possible events vanish (e.g., that of the choice of a certain point on a segment of a straight line or curve), as if those were impossible events. +∞is a lot of many different infinities hardly distinguishable.

In ancient times, it was counted as follows: 1, 2, many; the concept „many“ played the role of a lot.

Classical mathematics [1] cannot precisely measure different infinities at all. For example, any classical mathematical measure of each segment or interval on a straight line or a curve is independent of whether or not that includes its endpoints.

Bolzano [12] has stated his dissatisfaction with such circumstances and tried to do something in the particular case of a natural-number length.

Cantor [13] has introduced cardinal numbers to roughly discriminate very different infinities only. For example, the cardinal numbers of the segment [0, 1] and of the complete three-dimensional space are both equal to the same continuum cardinality c. The cause is that, in contrast to the real numbers, each infinite cardinal number absorbs all the less and even equal cardinal numbers.

Infinitely small infinitesimal numbers have been already introduced by Leibniz [14] and justified by Sikorski [15], Klaua [16], Robinson [17], and Conway [18] (see also [19-21]). But these do not allow to conveniently express each required amount, e.g., that probability of the choice of a certain natural number. Belonging and existence answers ("This is an infinitesimal number"; "There is such an infinitesimal number") bring no satisfaction because all the necessary operations should be also applicable to the appropriate real number generalization.

If there were no equality such as

i² = -1,

the situation with the imaginary numbers would be similar. It would be impossible to both determine specific solutions to many very important equations and operate on such numbers. Instead of this, one could say: "That is an imaginary number", "There is such an imaginary number".

The conclusion is obvious. It is an urgent problem, in a suitable extension of the real numbers, to exactly express all the infinitesimal, finite, infinite, and combined pure (dimensionless) amounts and conveniently operate on them.

2. Operations

Abstract. The system of operations in classical mathematics has gaps because it cannot mathematically model any concrete (mixed) physical magnitude (quantity with a measurement unit). Even the pure number operations in classical mathematics are considered to be at most countable. Even finite pure number operations in classical mathematics can have very narrow correct definition domains.

In classical mathematics [1], in each concrete (mixed) physical magnitude, e.g., 5 liter fuel, the operation unifying "5 L" and "fuel" is not obvious.

It is impossible to consider either "fuel multiplied by 5 liter" or, all the more, "5 liter multiplied by fuel". These both pure theoretical possibilities are not reasonable at all. And classical mathematics [1] cannot propose nothing else.

Hence for any concrete (mixed) physical magnitude (quantity with a measurement unit), there is no suitable mathematical model and no known suitable operation.

Nota bene: Multiplication is the evident operation between the number "5" and the measurement unit "L".

Further even the pure number operations in classical mathematics [1] are considered to be at most countable, which makes the range of mathematical models very narrow.

Such a finite pure number operation as raising a negative number to a power of 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 .

3. Sets

Abstract. The classical Cantor sets with either unit or zero quantities (0 and 1 are the only values of the indicator functions in the classical Cantor sets) 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.). 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. 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.

In classical mathematics [1], the Cantor set theory [1, 13, 22-25] is the basis of contemporary classicalon mathematics. The classical Cantor sets [1, 13, 22-25] with either unit or zero quantities (0 and 1 are the only values of the indicator functions in the classical Cantor sets [1]) of their possible elements may contain any object as an element either once or not at all, and its further repetitions are ignored. No other quantities of the existing elements are considered. For example, the following two sets are exactly equal to each other: the first one consists of a million of 1 Euro coins conditionally indistinguishable, and the second one consists of one 1 Euro coin. That is, a millionaire is equivalent to a poor person. Of course, in many cases, the Cantor set theory gives much healthier models than we often see. But frequently it does not work. And the fundamental conservation laws are broken. Hence this theory cannot simulate processes with holding these universal laws. No other quantities of the existing elements are considered also by Cantor set relations and operations [1, 13, 22-25]. Those set operations with absorption are only restrictedly invertible. The simplest equations

X ∪ A = B

and

X ∩ A = B

in X are solvable only by

A ⊆ B

and

A ⊇ B ,

respectively (uniquely by

A = ∅

and

A = B = U

(a universal set [1, 13, 22-25]), respectively). The equations

X ∪ A = B

and

X = B \ A

are equivalent by

A = ∅

only.

In a fuzzy set [26, 27], the membership function of each element may also lie strictly between these ultimate values 0 and 1 in the case of uncertainty only.

Element repetitions are taken into account in multisets [28] with any cardinal numbers as multiplicities and in ordered sets (tuples, sequences, vectors, permutations, arrangements, etc.) [1, 22, 23]. Unordered combinations with repetitions are considered in [23] to be no sets but equivalence classes of equipartite arrangements with repetitions. None of them can express and model many typical unordered objects collections (without structure), e.g., that of half an apple and a quarter pear. Even if 1/2 and 1/4 are pure numbers between 0 and 1 (the only values of the indicator functions in the classical Cantor sets [1]) like membership functions in fuzzy sets [26, 27], the fuzzy sets cannot model this unordered objects collection because it consists of exactly half an apple and a quarter pear and hence is nonfuzzy (exact, definite, well-defined, strictly defined).

In many unordered collections (for example, of bonds and coins), namely the quantity (multiplicity, number, amount) of each element is decisive and can be arbitrary, for example, 3.5 kg, 1 loaf (bread), - 45 € (Euro) by buying and those with opposite signs by selling.

In the unusually sincere book “The Paradoxes of the Infinite” [12], Bolzano was possibly the first who expressed his dissatisfaction with the fact that classical mathematics is powerless to quantitatively reflect many finite and even infinite changes in the infinite sets. He attempted to make something in the very special case of rectangles with the integral sides only. But all these attempts were later declared to be misbelief with mistakes…

4. Cardinalities

Abstract. 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.

In classical mathematics [1], the Cantor theory of cardinal numbers [1, 13, 24, 25] gave the first instrument of comparing and measuring distinct infinities. This tool remains the only in classical mathematics and is useful. However, it is low-sensitive. When you either add to an infinite set still the same one or divide an initial infinite set into two equivalent infinite sets and leave only one of them, their cardinalities remain the same and are equal to each other. Moreover, the unit interval, the entire three-dimensional infinite space, and even any spaces of countable dimensionalities have the same continuum cardinality. Infinity simply remains a heap of infinities completely distinct and extremely roughly (by cardinal numbers only and nothing more) divided into the classes. The two of them are especially important: the class of countable sets with their common cardinal number denote by aleph with index zero and the class of the sets of the power of continuum with their common cardinal number. The ancient numerical scale is recalled: one, two, many… Here the concept “many” unifies all the further natural numbers (positive integers) beginning with 3 without their separation.

5. Measures

Abstract. 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. There are no sensitive common measures for any even bounded sets of mixed dimensions.

In classical mathematics [1], each measure is sensitive only restrictedly, namely in the limits of the specific dimensionality, and even within them completely ignores even uncountable zero-measure changes.

Any measure of each segment or interval on a straight line or a curve is independent of whether or not that includes its endpoints.

In classical mathematics [1], there are no sensitive common measures for any even bounded sets of mixed dimensions, i.e., sets simultaneously including parts of different dimensions such as separate points, intervals, as well as bounded parts of surfaces and spaces.

Bolzano [12] has stated his dissatisfaction with such circumstances and tried to do something in the particular case of a natural-number length.

6. Probabilities

Abstract. The probabilities cannot discriminate impossible and some differently possible events. The probabilities of reasonable possible events can be nonexising at all.

In classical mathematics [1], real numbers having gaps between them cannot express not only unlimited, but also many limited quantities (e.g., the probability of selecting one given number from all natural numbers).

Let us assume that there are 10 balls with ciphers (digits) 0, 1, ... , 9, respectively, in a bag. Precisely one of the balls is picked out blindly (randomly, without any extrasensory abilities). What is the probability that the picked ball has namely a given (predefined) cipher (digit), for example, 7? The total number of all the possible outcomes is 10. By only one of them, the desired event occurs. That is why the desired probability by its classical definition is 1/10.

Let us now consider a more complicated problem. Imagine that we select precisely one number from countably many nonnegative integers 0, 1, 2, ... , 10, ... , 100, ... , 1000, ... with equal probability of selecting any of them. What is this probability or, equally, the probability that we have selected namely a given (predefined) number, for example, 7? Classical mathematics declares that this probability does not exist at all because of the following “proof”. Ad absurdum, suppose that this probability exists. Then it has to be either 0 or positive. If it is 0, then the total probability of selecting any of nonnegative integers whose total number is finite also wanishes. The same holds for all the nonnegative integers, which is proved via the corresponding limiting process. But the total probability of selecting any of all the nonnegative integers has to be precisely 1 as the probability of a certain event. Indeed, exactly one of all the nonnegative integers is selected. If the desired probability is, on the contrary, any positive number, then divide 1 by this number and take any nonnegative integer which is greater than this quotient. The so-called axiom of Archimedes provides that there exist infinitely many such nonnegative integers. For definity, take the least from these nonnegative integers. Then the total probability of selecting any of nonnegative integers from 0 to the taken nonnegative integer is greater than 1, which is impossible for any probability at all. Morover, in this case of any positive desired probability, the corresponding limiting process even gives plus infinity for the total probability of selecting any of all the nonnegative integers instead of 1 as the probability of this certain event. In this way, classical mathematics leads to the conclusion that desired probability does not exist at all.

Note that the Archimedes axiom [1] in mathematics is an invention, which is typical for entire mathematics itself, whereas the Archimedes law on a buoyant force [1] much better known is a law of nature fully objective and hence is a discovery, which is typical for natural sciences.

Classical mathematics also declares without any explanations that if the probability of selecting anyone of the elements of any uncountable set, for example, an interval, a straight line, a rectangle, a plane, of a space, is the same, then it vanishes, as if would be an impossible event.

But these and many other typical events are fully reasonable and possible and hence must have certain positive probabilities. And if classical mathematics cannot indicate them, then its real number system is clearly insufficient and has gaps.

It is possible to assume that such probabilities are indeterminate nonstandard numbers of Robinson [17].

Let us give the following analogy. It is necessary to solve a certain equation. How much benefit would we achieve due to the conclusion that the solutions to this equation are some undetermined imaginary numbers without their determination and clear indication? It is not difficult to guess the school grade for such an answer…

It occurs that in a whole series of key directions, by the level of thinking, modern classical mathematics corresponds to physics from the antique times to the 19th century, which also considered its atoms as indivisible. In the 20th and 21st centuries, physics slowly deepens their final division into the component parts. It requires such research monsters as the Large Hadron Collider. And physics is indeed quite foremost natural science…

7. Nonexistence of Contradictory Objects, Systems, and Models

Abstract. Classical mathematics intentionally avoids, ignores, and cannot (and possibly hence does not want to) adequately consider, model, express, measure, evaluate, and estimate many complications such as contradictions, infringements, damages, hindrances, obstacles, restrictions, mistakes, distortions, errors, information incompleteness, multivariant approach, etc.

In classical mathematics [1], contradictory objects, systems, and models are declared nonexisting at all and are completely ignored along with contradictory problems even if they are urgent. 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.

Nota bene: 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].

Therefore, the very fundamentals of classical advanced 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. General Estimation Theory. Transactions of the Ukrainian Glass Institute 1 (1994), p. 214-221 (both this article and a further mathematical monograph have been also translated from English into Japanese)

[3] Lev Gelimson. Basic New Mathematics. Drukar Publ., Sumy, 1995

[4] Lev Gelimson. Quantianalysis: Uninumbers, Quantioperations, Quantisets, and Multiquantities (now Uniquantities). Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 15-21

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

[6] Lev Gelimson. Providing helicopter fatigue strength: Flight conditions [Megamathematics]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Vol. II, Dalle Donne, C. (Ed.), Hamburg, 2005, p. 405-416

[7] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2009

[8] Lev Gelimson. Overmathematics: Principles, Theories, Methods, and Applications. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2009

[9] Lev Gelimson. Uniarithmetics, Quantialgebra, and Quantianalysis: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations. Mathematical Monograph. The “Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2010

[10] Lev Gelimson. Uniarithmetics, Quantianalysis, and Quantialgebra: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations (Essential). Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 11 (2011), 26

[11] Lev Gelimson. Science Unimathematical Test Fundamental Metasciences Systems. Monograph. The “Collegium” All World Academy of Sciences, Munich (Germany), 2011

[12] B. Bolzano. Paradoxien des Unendlichen. Bei C. H. Reclam Sen., Leipzig, 1851

[13] G. Cantor. Gesammelte Abhandlungen. Springer-Verlag, Berlin, 1932

[14] G. W. Leibniz. Sur les monades et le calcul infinitesimal, etc. Letter to Dangicourt, Sept. 11, 1716. In: Leibniz, G. W.: Opera Omnia (ed. by L. Dutens), Vol. 3 (1789), 499-502

[15] R. Sikorski. On an ordered algebraic field. Sci. Math. Phys. 41 (1950), 69-96

[16] D. Klaua. Zur Struktur der reellen Ordinalzahlen. Zeitschrift für math. Logik and Grundlagen der Math. 6 (1960), 279-302

[17] A. Robinson. Non-Standard Analysis. North-Holland, Amsterdam, 1966

[18] J. H. Conway. On Numbers and Games. Academic Press, London, 1976

[19] Real Numbers, Generalizations of the Reals, and Theories of Continua (ed. by Ph. Ehrlich). Kluwer Academic Publisher Group, Dordrecht, 1994

[20] Rechnen mit dem Unendlichen. Beiträge zur Entwicklung eines kontroversen Gegenstandes (ed. by D. D. Spalt). Birkhäuser, Basel, 1990

[21] Martin Davis. Applied Nonstandard Analysis. Wiley, New York, London, Sydney, Toronto, 1977

[22] G. A. Korn, T. M. Korn. Mathematical Handbook for Scientists and Engineers. McGraw-Hill, N.Y. etc., 1968

[23] I. N. Bronstein, K. A. Semendjajew. Taschenbuch der Mathematik. Frankfurt/M., 1989

[24] A. Fraenkel. Einleitung in die Mengenlehre. Berlin, 1928

[25] F. Hausdorff. Mengenlehre. Berlin, 1935

[26] L. A. Zadeh. Fuzzy sets. Information and Control 8 (1965), 338-353

[27] D. Dubois, H. Prade. Fuzzy Sets and Systems: Theory and Applications. New York, 1980

[28] W. Blizard. The development of multiset theory. Modern Logic 1 (1991), no. 4, 319-352