# Mathematics

##### Article

July 5, 2022

Mathematics, or arithmetic, is a special science in terms of its subject and methods, which examines the systems and structures, partly investigated by other sciences, and partly created (discovered or invented) as a result of the "internal" development of mathematics, and their abstract, jointly existing properties. In the past, it was defined as the "science of quantity and space" (that is, the science of numbers and geometric shapes), and from the beginning of the last century, mathematics was considered to be "the study of the abstract structures of set theory with a formal logical approach and the mathematical notation system based mostly on it examination". Today, not only the first, but also the second point of view is considered debatable and outdated by representatives of some philosophical and didactic currents. It is difficult to precisely define mathematics, and the question of its existence is even today, especially nowadays, a subject of debate, a live and not closed scientific problem, which is dealt with by the philosophy of mathematics (a field of philosophy, even an independent discipline). As long as there was an agreement, if not complete, on the activities, methods and concepts classified as mathematics (i.e. what is within the scope of mathematics and what is not) to a high degree and which hardly provoked disputes, then the subjects studied by mathematics about the ontological situation of concepts, their epistemological approach, about the nature and even the purpose of mathematics itself as an activity ("hard-line" natural or "soft" social science, perhaps art; empirical or rather normative, etc.), the most diverse relevant ideas exist in parallel with each other. Therefore, in the following, we will instead try to highlight some important, distinguishing features, some of which can be found in other sciences, but all together only in mathematics. The peculiarity of mathematics lies primarily in its special choice of topics, research areas and methods, language and notation system.

## High degree of abstraction and specialization

Even the simplest mathematical concepts, such as the concept of the number or the point, are the results of a high degree of abstraction, and historically almost always over the course of several centuries or millennia. During this process, looking at a set of things (objects, concepts), we abstract from the properties that are unimportant from the point of view of the investigation, and we only take into account certain prominent properties. Mathematically, a concept created as a result of an abstraction can be identified with the set of things that belong to the concept. In mathematics, there is often a conceptualization process called specialization. Logically, this corresponds to forming a subset from a concept, and in set theory, a subset from a set. This is how we get, for example, "Belgian Shepherd" from the concept of "dog", "halogen element" from the concept of "chemical element", "even" or "odd" from the concept of "number" (whole number)

## Special Methods

Its methods are also very typical of this science, the main ones being the deductive or axiomatic knowledge acquisition system based on the science of mathematical logic (the so-called axiomatic-deductive method) and a special mathematical language and notation system (primarily based on the symbolism and language of set theory).

## Special language and symbolism

Since mathematics often works with concepts and methods that occur only sporadically in "real" life and other sciences, a special technical language was needed,