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A (rough) description of some class of events of the outside world, expressed using mathematical symbolism. A mathematical model is a powerful tool for understanding the outside world and for prediction and control. The analysis of a mathematical model allows the essence of a phenomenon to be penetrated. The process of mathematical modelling, that is, the study of phenomena with the aid of mathematical models, can be divided into four stages.

The first stage is the statement of the laws relating the basic objects of the model. This stage requires a broad knowledge of the facts relating to the phenomenon and a high penetration into their interconnections. This stage is completed with the description, in mathematical terms, of the formulated qualitative representations of the connections between the objects of the model.

The second stage is the investigation of the mathematical problems to which the mathematical model leads. The fundamental question here is the solution of the direct problem, that is, as a result of analysis of the model to obtain output data (theoretical consequences), which are then compared with the results of observation of the phenomenon. In this stage the mathematical apparatus necessary for the analysis of the mathematical model and the computational techniques — the powerful means for obtaining quantitative output of information as a result of the solution of complex mathematical problems — assume a major role. Frequently mathematical problems arising on the basis of various mathematical models turn out to be identical (for example, the basic problem of [[Linear programming|linear programming]] reflects various situations in nature). This provides a basis for considering these typical mathematical problems as independent objects abstracted from the phenomena.

The third stage is the clarification of whether the adopted (hypothetical) model satisfies the practical criterion, that is, whether the observed results agree with theoretical consequences of the model within the limits of accuracy of the observations. If the model was fully determined, all its parameters being given, then determination of the derivation of the theoretical consequences from the observations gives the solution to the direct problem with a posteriori estimates of the deviation. If the deviation lies outside the limits of precision of the observations, then the model cannot be accepted. Frequently, in the construction of models certain characteristics are left undetermined. Problems in which the characteristics of the model (parametric, functional) are determined so that the output information is comparable, within the limits of precision of the observations, with results of observations of the phenomenon, are called inverse problems. If a mathematical model is such that there is no choice of characteristic which satisfies these conditions, then the model is useless for the investigation of the phenomenon. The application of a practical criterion to appraise a mathematical model allows one to draw conclusions on the validity of the assumptions underlying the (hypothetical) model being studied. This is the only method of studying phenomena of the macro- and micro-world which are not directly accessible.

The fourth stage is the a posteriori analysis of the model in conjunction with the observation data of the phenomenon, and the updating of the model. In the process of development of science and technology, phenomenological data become more and more precise and the time comes when the output obtained on the basis of an accepted mathematical model does not correspond to knowledge of the phenomenon. Therefore the need arises to construct a new, more precise, mathematical model.

A typical example illustrating the characteristic stages in the construction of a mathematical model is the model of the solar system. Observation of the stars in the sky began in early Antiquity. A primary analysis of these observations allowed one to pick out the planets from the whole variety of celestial bodies. Thus, the first stage was the selection of the objects of study. The second stage was the determination of regularity in their motions (generally, the definition of the objects and their interconnections are the starting point,  "axioms" , of the model). The model of the solar system during this development passed through a number of successive refinements. The first was the Ptolemeus model (2nd century B.C.), starting from the position that the planets and the Sun moved around the Earth (the geocentric model), and when this motion had been described by rules (formulas), these became increasingly complicated with the increase of observations.

The development of navigation raised, for astronomy, new requirements for precision in observation. N. Copernicus in 1543 suggested a fundamentally new basis for the laws of motion of planets by assuming that the planets rotate around the Sun in circles (the heliocentric system). This was a qualitatively (but not mathematically) new model of the solar system. However, the parameters of the system (the radii of the circles and the angular velocities of the motions) could not be measured, which resulted in quantitative conclusions of the theory not in proper correspondence with observations, so Copernicus was forced to introduce amendments to the motions of the planets by circles (epicycles).

The next stage in the development of the model of the solar system was the research by J. Kepler (1571–1630), in which the laws of planetary motion were formulated. The aims of Copernicus and Kepler were to give a kinematic description of the motion of each individual planet without touching upon the reasons for these motions.

A principally new stage was the work of I. Newton, who proposed, in the second half of the 17th century, a dynamic model of the solar systems based on the law of universal gravitation (cf. [[Newton laws of mechanics|Newton laws of mechanics]]). The dynamic model was consistent with the kinematic model suggested by Kepler, since Kepler's laws followed from the dynamical system of two bodies,  "Sun–planet" .

In the 1840's, conclusions of the dynamic models, the objects of which were the visible planets, were found in contradiction with the observations collected at that time. Namely, the observed motion of the planet Uranus deviated from the theoretically calculated motion. U. le Verrier in 1846 expanded the system of observed planets with a new hypothetical planet, named by him Neptune, and, by using the new model for the solar system, determined the mass and the law of motion of the new planet in order that in the new system the contradiction in the motion of Uranus was removed. The planet Neptune was discovered at the place stated by le Verrier. By a similar means, using the divergence of Neptune, the planet Pluto was discovered in 1930.

The method of mathematical modelling, reducing the investigation of the phenomena of the outside world to mathematical problems, occupies a leading position among methods of research, particularly in connection with the appearance of electronic computers. It allows one to design new technical means of working in optimal regimes for the solution of complex scientific and technical problems and to predict new phenomena. Mathematical models have been shown to be an important means of control. They are applied in very diverse domains of knowledge and have become a necessary tool in economic planning and an important element in automatic control systems (cf. [[Automatic control theory|Automatic control theory]]).



====Comments====
For models in logic (i.e. models of axiomatic systems), cf. [[Model (in logic)|Model (in logic)]] and [[Model theory|Model theory]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bochner,   "The role of mathematics in the rise of science" , Princeton Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.S. Kuhn,   "The Copernican revolution" , Cambridge, Mass.  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Nagel,   "The structure of science" , London  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Rosenblueth,   N. Wiener,   "The role of models in science"  ''Philosophy of Science'' , '''12'''  (1949)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.A. Bender,   "An introduction to mathematical modelling" , Wiley  (1978)</TD></TR></table>