Physics was purely a descriptive form of science not so long ago — just a few ages ago. However there has been a gradual change shift from collecting and describing facts to examination of connections between them. As systematisation of knowledge and occurrence of general concepts went on, the role of mathematical methods in physics increased as well. First of all, this mathematisation process began in the mechanics. Logically coherent system of mechanics was created by Newton and described in the book “The Mathematical Principles of Natural Philosophy” (1687). This book played the same role for mechanical development as the illustrious Euclid’s Elements in geometry. Newton went all the world system through mathematical analysis, observing the movement of movement of terrestrial and celestial bodies. His book is structured along the lines of mathematical research: it begins with definitions and postulates, from which theorems and consequences are derived. Newton’s work has long remained a model for the construction of physical discipline. Because of that, mathematical basics of mechanics have been formed already in the 17th century. Much later common mathematical methods penetrated the electromagnetic phenomena. Just around 100 years ago J.C. Maxwell developed mathematical theory of electromagnetism. Theoretical physics became a separate scientific discipline just around 100 years ago. M. Planck writes in his autobiography that in the 1880s theoretical physics did not exist as a thing. When in 1989 he became a professor of theoretical physics at Berlin University, this position was viewed as something unusual by his colleagues. ‘ I was then, probably the only theoretician… ‘— Planck remembers. However, just after a short period of time and L. Boltzmann could have confidently said that ‘theory conquered the world’. Mathematical methods penetrated not only into mechanics and electrodynamics, but also into the kinetic theory of gases, thermodynamics, and then into atomic physics. Mathematics has made it possible to predict radio waves, the existence of new planets, elementary particles and their basic properties. Mathematics not just influenced physics or astronomy but was also under their influence itself. Problems of mechanics stimulated the creation of differential and integral calculus, and these same problems stimulated the development of variational calculus. Problems of heat propagation, vibrations of strings and membranes, problems of fluid and gas mechanics contributed to the development of the theory of partial differential equations, the theory of trigonometric series, and some other areas of mathematics. For a long period of time physics and astronomy have been the primary source of mathematical problems and key areas where the power of new mathematical methods has been tested. However, in recent years things have significantly changed. We are the eyewitnesses of the fact that mathematical methods integrate economics, sociology, linguistics, biology and so on. Such terms as mathematical linguistics, mathematical economics, and mathematical biology emerged and entered into common use. What is the reason behind so powerful invasion into areas, that were quite recently so remote from each other, biology in particular? And why is this happening right now and not, let’s say, 100 years ago? Biology has long been a descriptive science, a collection of more or less systematized results of observations and experiments. As this factual material accumulated, deep connections began to be discovered between phenomena that had previously seemed isolated. For example, the work of the cell’s genetic apparatus turned out to be connected with the processes of protein synthesis, and at the same time with the processes of organism formation. On the other hand, genetic patterns, along with the theory of natural selection, formed the basis of modern ideas about evolution. Thus, metabolism, heredity, morphogenesis, and evolution turned out to be closely connected, and biology came closer to understanding the mechanisms underlying these connections. In other wors, nowadays huge success has been achieved in terms of understanding primary biologic patterns. This success raised interest in biology’s common problems, reinforced the aspiration to define the common principles of biological systems operations, to understand the essence of life. All of these became precursors of theoretical biology and also the intervention of mathematical methods into biology, which played a colossal role in development of theoretical physics, mechanics, and astronomy. Apart from the primary reason behind the creation of theoretical biology, there are some other circumstances behind penetration of mathematics into biology. One of which is the development of new disciplines, lying on the borderline of different sciences. In biology new sections started to appear bordering those sciences, where mathematics is applied for ages and quite successfully. These are primarily biophysics, biochemistry, and molecular biology. The biological problems studied by these disciplines are often able to be formulated as chemical physics problems (even though quite specific ones). Another developing path of modern biology is researching control processes in living organisms. This path has not been shown as a separate discipline yet, but it is heavily developing at different stages: Cytologists examine regulatory processes in individual cells, while physiologists study the structure and functions of the nervous system, the main control system of the body, ecologists study regulation in biological communities, etc. Problems of control, transmission and processing of information, learning, memory, etc. are currently attracting the attention of not only biologists, but also specialists in the field of technology. One of the vital tasks of modern technology is the creation of fast responding automatic means of control. Theoretical problems connected with control attracted the attention of many mathematicians. Assuming that some parts of process are likewise, hence mathematical methods developed for technology application could be applicable to biology as well. If biologists are aiming to find in the theories of information, automatic regulation and other technical disciplines ideas and methods suitable for researching biological control processes, engineers, on the other hand, are striving to find new principles, which would be possible to use in technologies. Actually, considering that millions of years of evolution are supposed to choose the optimal options, engineers are looking for ways to use these “natural discoveries” in computing, control systems, etc. This path, named Bionics, attracted to biology people with background in physics and mathematics. Moreover, a great number of such professionals came into biology, because of the new equipment and new ways of research. In a modern biological laboratory, amplifiers, oscilloscopes, electron microscopes, ultracentrifuges, and so on became nothing out of the ordinary. As a result, people, for whom mathematics is a familiar research technique became working for biology. Penultimately, since the development of borderline sciences (biophysics, biochemistry, bionics), occurrence of closely related paths in both biology and technology (control problems), and also the development of engineering and technical methods for studying biological objects, biologists are now working side by side with physicists, chemists, engineers and mathematicians. Finally, the development of biology itself has made natural and crucial a vast integration of mathematical methods. On the other hand, as a form of development of modern mathematics, new paths started to appear, connected with complicated system research and significantly increasing the possibilities of application of modern methods into biology.   Source: S. V. Fomin, M. B. Berkinblit: Mathematical problems in biology (In Russian).