Mathematics is the science of structures, procedures and relationships, which historically has developed on the basis of the operations of counting, measuring and describing shapes of real objects. Mathematical objects are created by the idealization of real property or other mathematical objects and write these properties in a formal language. Mathematics is not a natural science, but it is widely used in them as to the exact wording of their content, and to obtain new results. Mathematics is the basic science; it is a language for other sciences, which provides the interconnection. Mathematics papers differ from the papers of humanity sciences because of the nature of the mathematics.

Mathematics studies the imaginary, ideal objects and relations between them using the formal language. In general, the mathematical concepts and theorems do not necessarily match anything in the physical world. The main task of applied branch of mathematics is to create a mathematical model that adequately investigated the real object. The problem of maths theorist is to provide a sufficient set of convenient tools for achieving this goal.

The content of mathematics can be defined as a system of mathematical models and tools for creating them. Object model takes into account not all of its features, but only the most necessary for the purposes of the study (idealized). For example, studying the physical properties of orange, we can abstract from its color and flavor, and submit it (even if not perfectly accurate) as the ball. If we need to understand how many oranges will turn out, if we to add together two and three - that can be abstracted and the shape, leaving the model has only one characteristic - the number. Abstraction and establishing connections between objects in a general way is one of the main areas of mathematical creativity.

The question of the nature and foundations of mathematics has been discussed since Plato. Starting from the XX century there has been a comparative agreement on an issue that must be considered a rigorous mathematical proof, but there has been no agreement to any understanding of what mathematics considers to originally be true. This implies differences both in terms of axioms and relationships of branches of mathematics, and in the choice of logical systems, which should be used in the proofs.

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