an approximate description of some class of phenomena of the external world expressed by means of mathematical symbolism. A mathematical model is a powerful method of understanding the external world as well as of prediction and control. Analysis of a mathematical model allows us to penetrate the essence of the phenomena under study. The process of mathematical modeling, that is, the study of a phenomenon using a mathematical model, can be divided into four stages.

The first stage consists in formulating the laws that relate the principal objects of the model. This stage requires a broad knowledge of the facts pertaining to the given phenomena and a deep understanding of the interrelations between the phenomena. The final step of this stage is the notation in mathematical terms of formulated qualitative conceptions of the relations among the objects of the model.

The second stage consists in investigating the mathematical problems that the mathematical model leads to. Here the main question involves the solution of the direct problem, that is, obtaining from an analysis of the model various output data (theoretical conclusions) for further comparison with observations of the given phenomena. At this stage, the mathematical apparatus required for analyzing the mathematical model assumes an important role as does computer technology, a powerful tool for obtaining quantitative output information through the solution of complex mathematical problems. Often, the mathematical problems that arise in connection with mathematical models for different phenomena are identical; for example, the fundamental problem of linear programming covers quite different kinds of situations. This justifies the examination of such a typical mathematical problem as an independent subject of study abstracted from the phenomena under consideration.

The third stage consists in ascertaining whether or not the given hypothetical model satisfies the criterion of practice, that is, determining whether the results of observations agree with the theoretical consequences of the model within the limits of observational accuracy. If the model is completely defined, that is, all its parameters are given, then a determination of the deviations of the theoretical conclusions from the observations provides a solution of the direct problem as well as an estimate of the deviations. If the deviations exceed the limits of the observational accuracy, the model cannot be adopted. The setting up of a model often leaves some of the model’s characteristics undefined.

Inverse problems are problems in which the characteristics (parametric or functional) of a model are defined such that the output information can be compared within the limits of observational accuracy with the results of observations of the phenomena under study. If these conditions cannot be satisfied for a given mathematical model, no matter how the characteristics are selected, the model is unsuitable for investigating the phenomena in question. The use of the practice criterion in evaluating a mathematical model allows us to determine whether the assumptions on which the (hypothetical) model being studied is based are correct. This method is the only way of studying phenomena of the macroworld or microworld that are not directly accessible to us.

The fourth stage consists in analyzing the model in conjunction with new data on the given phenomena and updating the model. Data on the phenomena being studied are continuously being refined in the course of the development of science and technology, and a point is reached when conclusions that can be drawn from existing mathematical models do not correspond to our knowledge of the phenomenon. Thus, it becomes necessary to set up a new and more perfect mathematical model.

The model of the solar system illustrates the characteristic stages in setting up a mathematical model. Observations of the heavens were begun in remote antiquity. The first analysis of these observations made it possible to separate the planets from the rest of the heavenly bodies. Thus, the first step consisted in isolating the objects of study. The second step was the determination of the regularities in their motions. (In general, determinations of objects and their interrelations are the initial assumptions, or “axioms,” of a hypothetical model.) Models of the solar system underwent a number of successive developments. The first such development was Ptolemy’s model (second century B.C.), which proceeded from the assumption that the planets and the sun revolve around the earth (geocentric model) and which described these motions by means of rules, or formulas, that became increasingly complex as observations accumulated.

The expansion of seafaring brought about a need for better observational accuracy in astronomy. In 1543, N. Copernicus proposed a basically new foundation for the laws of planetary motion that assumed that the planets revolve around the sun in circular orbits (heliocentric system). This constituted a qualitatively new, but not mathematical, model of the solar system. However, there existed no parameters for the system (radii of the circles and angular velocities) that would make the quantitative conclusions of the theory show the necessary correspondence with observations, and therefore Copernicus was forced to introduce corrections (epicycles) into the motions of the planets along circular paths.

The next step in the development of the model of the solar system was taken by J. Kepler (early 17th century), who formulated the laws of planetary motions. The conclusions of Copernicus and Kepler provided a kinematic description of the motion of each planet by itself but did not yet touch upon the reasons for these motions.

The studies of I. Newton constituted a basically new step. In the second half of the 17th century Newton proposed a dynamic model of the solar system based on the law of universal gravitation. The dynamic model agrees with the kinematic model proposed by Kepler since Kepler’s laws follow from the dynamic two-body “sun-planet” system.

By the 1840’s the conclusions of the dynamic model, whose objects were the visible planets, came to contradict the observations that had accumulated by that time. Specifically, the observed motion of Uranus deviated from the theoretically calculated motion. In 1846, U. Leverrier expanded the system of observed planets with a new hypothetical planet he called Neptune and, using a new model of the solar system, determined the mass and law of motion of the new planet in such a way that in the new system the contradiction in the motion of Uranus was removed. The planet Neptune was discovered where Leverrier had predicted. The planet Pluto was discovered in 1930 by a similar method using discrepancies between the theoretical and observed trajectories of Neptune.

The method of mathematical modeling, which reduces the study of phenomena of the external world to mathematical exercises, occupies a central place among other methods of investigation, particularly since the advent of the electronic computer. Computers make it possible to devise new technological facilities operating under optimal conditions for the solution of complex scientific and technological problems; they also make it possible to predict new phenomena. Mathematical models themselves have proved to be an important means of control. They are used in the most varied branches of knowledge and have become a necessary apparatus in economic planning; they are also an important element in automated control systems.