by [Sergei KURDYUMOV], RAS Corresponding Member, Head of Department, RAS Institute of Applied Mathematics named after M. V. Keldysh
In 2004 one of the Moscow publishing houses brought out a book called Asymptotic Mathematics and Synergetics. Its authors - Prof. Igor Andrianov, Dr. Sc. (Phys. & Math.), Rem Barantsev, Dr. Sc. (Phys. & Math.), and Leonid Manevich, Dr. Sc. (Tech.) made an attempt to describe in a popular form the essentials of these new branches of knowledge.
Biographers studying the legacy of Albert Einstein, Nobel Prizewinner for 1921 and Honorary Foreign Member of the USSR Academy of Sciences since 1926, quote him as saying that sciences are divided into physics and stamp collection. If so, what did the outstanding scholar and philosopher have in mind emphasizing that particular discipline?
And one can also recall the words of Galileo Galilei (1564 - 1642), one of the founding fathers of natural sciences. He said that-be it white or red, bitter or sweet, sounding or silent, with pleasant or unpleasant smell - all these are mere "names" for different impacts upon our sensory organs. He said he would never expect from external objects anything else than data on their size, shape, numbers and more or less rapid movements in order to explain the origin of sensations of taste, smell and sounds. He said that if people had no ears, tongues and noses they would have been left with only shapes, numbers and movements which, taken outside of a living being, are nothing but empty names of definitions. In other words, the key principle of idealization suggested by the great Italian scholar for physics-imaginative building of notions about objects, processes and phenomena which do not exist in reality, but have prototypes (such as "point", "absolutely solid, hard body"). This makes it possible to formulate laws, build abstract models of real events. Using this approach Galilei studied movements of bodies. He pointed out that in order to examine this problem from a scholarly angle one
has to cast aside all the aforesaid difficulties (air resistance, friction, etc.) and, having formulated and proved theorems for cases when there is no friction, use them with the limitations suggested by experience. Incidentally, the principle of idealization does not belong to man's "inborn conceptions" and one has to learn it. In our view, however, no due attention is given to such basic concepts either in schools or in colleges. Our outstanding physicist, Acad. Leonid Mandelstam (from 1929) stressed years ago that problems of idealization should take a fundamental place in any teachings of physics - at school and in universities. He said that even a schoolboy must realize that dealing with any physics theory we are dealing with ideal models of actual things and processes.
But what about passing from them to things in reality? The consistent search for solutions of this problem pursued by researchers produced what is called the method of perturbations. Another outstanding man of science - the French astronomer, mathematician and physicist de Laplace (1749 - 1827) (from 1802 Foreign Honorary Member of the St. Petersburg Academy) once said that the simplest method of analysis of different perturbations (in movements of planets. - S. K.) is to imagine a planet in an elliptic orbit with smoothly changing elements and imagine at the same time that the real planet is "oscillating" around that imaginary line along a very small trajectory whose properties depend upon its periodic perturbations.
Further studies proved that the process of searching for more precise definitions can be continued. There appeared the term "asymptotics". As one can recall, in the school course of mathematics we already came across it-straight lines endlessly approaching certain curves when the independent variable X tends to infinity. And that means that, starting with certain values of X, curves can be replaced with asymptotes. In a similar way corresponding formulas describe with maximum accuracy phenomena with certain parameters equal to zero or infinity (limiting cases). Searching for these is one of the central tasks of science.
One of the founding fathers of asymptotic mathematics Acad. Poincare (from 1895 Foreign Corresponding member of the St. Petersburg Academy) wrote that scientists had been searching for them in two extreme areas: in the areas of the infinitely large and infinitely small. And they were found by an astronomer, because distances between heavenly bodies are vast, so vast that each of the luminaries looks like a dot; they are so vast that distinctions are overlooked because a dot is simpler than a body which has its fonm and quality. And the physicist was looking for an elementary phenomenon, dividing in his mind this body into infinitely small squares because the conditions of a problem involving slow and continued changes, when we pass from one point of the body to another, can be regarded as permanent within the limits of each of these small cubes or squares.
Development of asymptotic mathematics has led us to the understanding that an appropriate description is not only a convenient "tool" of analysis of nature, but is of a fundamental importance itself. This brought to life a branch of knowledge called asymptotology which is now in the stage of establishment. And it is probably only now that we understand the aforesaid statement of Einstein that in a stamp collection everything is equally important. But in science one has to know what can be ignored and how its importance can later be estimated. And are the "summits" of our scientific achievements the "final word" in the studies of nature? The Canadian pathologist who offered in 1936 his definition of "stress", Prof. Hans Seglier said that classical art, like photography, had insisted on the principle of a detailed image, whereas modern art is trying, abstracting itself from details, to operate with symbols, thus stressing the most essential elements in an object. Both of these principles are represented in science. Today preference is certainly given to what one can call "penetration" inside an object increasing the accuracy of the available instruments. This method is extremely effective, but in the unrestrained race for details one can lose sight of the whole.
Today there is every reason to say that we are entering an epoch of synthesis of sciences offering a general view of the world. A few decades ago it seemed that this role will be assumed by cybernetics. But the implacable judges-time and practical experience-have passed a different verdict. Passing a "durability test" is synergetics which studies the phenomena of self-organization. They have been known for quite some time although it was quite recently that there has appeared an adequate mathematical apparatus which makes it possible to describe effects of this kind. The thing is that they are of nonlinear nature and the ongoing "nonlinear revolution"-serious changes in the methods of solving problems closely connected with asymptotology - has made it possible for synergetics to make a convincing claim to the right to existence.
Such are the basic problems touched upon in the book "Asymptotic Mathematics and Synergism" published in the series "Synergism: from the Past to the Future", chief editor Georgiy Malinetsky, Dr. Sc. (Phys. & Math.). Publications in the series acquaint the reader with the key notions of the new branch of knowledge (nonlinearity, openness, instability, asymptomaticity), classical works on this subject and new approaches illustrated on concrete examples. Publications in the series are going on and we are in a position to keep an eye on the latest ideas of synergetics and keep abreast with the latest achievements of modern science.
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