Finite math

A field of mathematics that studies the properties of structures of finite nature that arise in mathematics and their applications. Such finite structures may include, for example, finite groups, finite graphs, and certain mathematical models of information converters, finite automata, Turing machines, etc. Sometimes the topic of finite mathematics is extended to include arbitrary discrete structures, such as certain algebraic systems, infinite graphs, certain types of computer systems, and modular automata.

The resulting discipline is called discrete mathematics and is identified with finite mathematics. The term "discrete analysis" is sometimes used synonymously with the terms "finite mathematics" and "discrete mathematics". In the following, the term "final mathematics" is used in the broad sense that includes discrete mathematics.

Unlike finite mathematics, classical mathematics is primarily concerned with the study of continuous objects. Whether we use classical mathematics or finite mathematics as a means of investigation depends on the type of problem being investigated and, consequently, on whether the model of the particular phenomenon is discrete or continuous. For example, in the problem of finding the mass of a radioactive substance at a given time with a certain degree of precision, we may regard the process of mass change during radioactive decay as continuous, when in fact it is known to be discrete. 

The division of mathematics into classical mathematics and discrete mathematics is quite arbitrary, because on the one hand there is a considerable interchange of ideas and methods between them and, on the other hand, the need often arises to study models that simultaneously possess both discrete. and continuous properties. It should also be noted that there are areas of mathematics that use discrete mathematical methods to study continuous models (for example, algebraic geometry), and on the contrary, the methods and ways of formulating problems typical of classical analysis usually use discrete structures (for example , asymptotic problems in number theory). These examples indicate the clear overlap between classical and discrete mathematics.

Finite mathematics represents an important trend in mathematics. It is possible to distinguish the typical topic of studies, methods and problems, the nature of which is largely determined by the need, characteristic of finite math, to reject the basic concepts of classical mathematics -limit and continuity- and by the fact that The powerful classical mathematical methods are often of little use in many finite mathematical problems. In addition to delimiting finite mathematics by specifying its object, it is also possible to define it by specifying its subdivisions.

These include combinatorial analysis, graph theory, coding theory, and functional systems theory. In these terms, finite mathematics represents the study of finite structures. In less restrictive terms, finite mathematics includes entire branches of mathematics, such as mathematical logic, as well as parts of these branches, such as number theory, algebra, computer science, and discrete probability.

However, the latter mathematics reached its greatest development in relation to practical problems, which were the source of the new science of cybernetics and its theoretical counterpart - mathematical cybernetics. Human practical activity confronts cybernetics with a wide variety of problems. Mathematical cybernetics studies these problems from the point of view of mathematics. It is also a rich source of ideas and problems for finite mathematics and has introduced completely new trends in this field.

In addition to those already mentioned, finite mathematics has a number of unique properties. Thus, in addition to the existing problems encountered through mathematics, infinite mathematics deals with problems related to algorithmic solubility and the construction of specific solution algorithms, issues of finite mathematics. Another unique feature of finite mathematics is the fact that it was essentially the first mathematical discipline to demonstrate the need for intensive study of discrete multiple extreme problems common in mathematical cybernetics. Classical mathematical methods for finding extremes rely heavily on smooth functions and have proven ineffective on these problems.

A unique feature of finite mathematics associated with problems with finite structures is that solution algorithms usually exist for many of these problems, whereas a complete solution of the problem in classical mathematics is often only possible under extremely strict constraints. Examples of such problems are the aforementioned problems with chess strategies and the minimization of Boolean functions. An example of such an algorithm is to examine all possible alternatives. Such algorithms are very cumbersome and of little practical use. If you want to know more about this, read this post here.

In this sense, new questions arise about the conditions that limit the number of alternatives and lead to the reduction of individual problems, characterized by specific parameter values, to a general problem, characterized by an infinite series of parameter values. Other problems arise in connection with the introduction of constraints on the solution methods that are natural for this class of problem. The formulation of these questions and the development of the relevant techniques are carried out for the specific models offered by the different branches of mathematics.