Кузнецов Дмитрий Юрьевич: другие произведения.

Why mathematics?(old)

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Поиск утраченного смысла. Загадка Лукоморья
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  • Аннотация:
    The relation between politics and mathematics is suggested. Development of mathematics is proposed as necessary condition of stability and prosperity of civilized countries

Governments, destructive parties and mathematics.

The government does not need mathematics for itself. At least, the governments do not need the research in mathematics; the funding of the defense projects sometimes is considered as more important. But the civilized society needs mathematics both for the political stability and the advances in science and technology.
From the first look, the enforcement of the national police and the national army could provide the required stability of the society; indeed, it works during a period comparable with the lifetime of one generation. On the other hand, the societies concentrated on self-defense may loss the internal stability. The instability may come from the destructive parties: communists, chauvinists, religious fanatics, and all kinds of mafias without any ideologic bases (criminal groups of killers, racketeers, robbers, frauders, deceptors etc.). At the lack of the mathematical education of the people, the destructive parties may become popular with demagogic propaganda, and the federal forces become non-efficient.
The people with poor mathematical education cannot analyze the statements critically, and cannot distinguish the information from the propaganda. In order to get the civilized society, it is not sufficient to spread the knowledge of economics, history, literature. The historical knowledge is not appropriate basement for the education, because the history is written by the victors. The education based on the historical science, may lead to the concept "Use any ways to reach the power, because en fin the strongest becomes winner, and gets the ability to justify his way to the superior power." This danger is analyzed in the novel The crime and the punishment by F.Dostoevsky but was not recognized by his contemporaries, and he could not prevent Russia from the crash in 1917.
The logic preserves the knowledge of history and literature from substitution by the ideologic dogmas and cliches. The principle To teach people how to think rather than what to think is not new. The ability to think, the robust logic comes from mathematics; it provides the ability to analyze the statements critically. This ability is important factor of the stability of the society.

Common errors

Our century is characterized by the wide spreading of various softwares, that does not require any logic from the user. The high level languages such as Maple and Mathematica allow to perform some analytic calculus without to know mathematics. The specific simulators allow to optimize various devices, from opto-electronic circuits to cars and aircrafts, and the designer has no need to know physics. However, such ignorance may lead to anecdotic results; for example, the water pump with efficiency larger than unity, or an optical medium that violates the Second Law of Thermodinamics, and so on. In the most of cases, the researchers, who were caught on such "results" do not even recognize their guilt: "This was failure of the spectrometer. It did not correct the data for reabsorption"; or "This is fault of the simulation software. It does not take into account the background loss", and so on, in the same manner, as the practising physicians do not need to know physics.
One student wrote the wrong result in her thesis. The adviser looked at the formula and told: there is mistake here. The student upset:"I took it from the Table of Integrals!". (There was misprint in the Table of Integrals.)
One colleague had constructed the "operator of phase" for quantum optics. He used the formula of differentiation of the logarithm and the contour integration. He considered all the cuts of the range of holomorphism of the logatirhmic functions as some mathematical abuse; this lead him to the completely wrong statements.
Several researchers, estimating some parameter, in serious discuss the "probability, that the parameter has value within the specified interval". (Sometimes, such interpretation allows the reasonable estimates; however, it is completely wrong and easy leads to errors.)
The list of the common mistakes, caused by the ignorance in mathematics, could be long. The most of such errors could be easily avoided with the basic mathematical education.

Level of knowledge

As soon as the people begun to believe the politicians, governors, instead of to control their activity, the politicians and governors come to organize the crimes disasters to make the political capital fighting them, instead of to prevent them; they begin to alter the information about their faults instead of to correct the faults.
As soon the physicists and engineers begin to believe the software and the adopted manuals on mathematics, instead of to go through the deduction, the vulgar interpretations spread; the authors copy the formulas from previously published works without to check them; and this leads to the degradation of the contents. The notations become fussy; new misprints appear, and the researchers becomes like maintainers of the mechanical piano, instead of to be a like musicians and composers.
It does not worth to repeat the formula E=mc2 after Einstein, if the sayer does not understand how to use it, for example, to calculate the possible exit of energy in the nuclear reaction on the base of knowledge of the atomic mass of the isotopes involved.
It is not worth to repeat "The Dow Jones goes up, hence our leader does well", if the sayer does not know what is "dow-jones" or how do politicians provide the temporal growth of the economical criteria just before the election.
Mathematics provides the ability to distinguish the known things from unknown ones. The citizen should be able to say:"I do not know the meaning of the formula by Einstein", or "I do not know the meaning of the criterion by Dow Jones", or "I do not know who was aggressor in that conflict", if this is indeed the case. Such a honesty is important factor of both, the political stability and the efficient advance of the technology.

Modern approach

One of problems of the deep mathematical education is the lack of time of students. This is common argument against the teaching of the fundamental mathematics in non-mathematical departments.
This difficulty is artificial. Actually, at schools, the students learn the history of mathematics instead of namely mathematics; beginning from the antique epochs when π was believed to be "an approximate number".
The mathematics by itself is not difficult. It can be based on few simple axioms of logics and arithmetics. Then, one can construct the real numbers, the functions, Euclid space, and deduce the Euclid's axioms instead of to memorize them. The same applies to the trigonometric formulas, formulas for binomial and many other staff, that actually have no need to be remembered. The students should get the ability to deduce quickly the formula they need, and to forget it as soon as they do not need it.
While, instead of to get such a universal ability, which would be useful for new (perhaps, still unknown) formulas, that may be required in this century, the students try to remember the formulas and data that were useful in the past century. I consider this as the main defect of the education at this planet.

The governments of underdeveloped countries are not interested in the improvement of the mathematical education. First, usually, the rulers of such countries are not so honest with their own tribe. With basic knowledge of logic, the people will see faults, internal contradictions and the misinformation in the official propaganda. Second, the impact of Mathematics in physics and technology does not improve much the economics based on the export of the prime profucts (oil, wood, minerals) and import of everything else.

The civilized countries should be interested in the update of the mathematical education; the city, the university, the department gets an advantage, being first to change from the historical way of education of mathematics to the sequential.
From my side, I'll be glad to make all the efforts to realize the sequential approach in the mathematical education. The scheme of such educational program is available at http://www.ils.uec.ac.jp/~dima/2009teachma.html

Copyleft 2009 by Dmitrii Kouznetsov. This text can be copied, distributed and modified under the CC-BY-SA copyright license. Attribute http://www.ils.uec.ac.jp/~dima/2009matica.html
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