What is Mathematics?
This may sound as a trivial question, but the answer is much more complex than that, so every study of mathematics should begin with defining what, in fact, it is.
The very word “mathematics” is of a Greek origin and it is very symbolic, for it was in ancient Greece where it got its more or less modern meaning. The word from which it originates means simply “science” or “learning”, but already in ancient times it has developed and expanded towards its modern meaning.
Mathematics is one of the purely abstract sciences, for it studies numbers, quantities, changes, structures, patterns – the things that exist only as far as there are people to perceive them and created only by people’s abstractions. Even among the mathematicians themselves there is no unanimity about whether the concepts they study actually exist. Nevertheless, mathematics as a science developed from primitive activities like counting or measurement, observations of objects in reality and so on, and only in course of time moved into the sphere of pure abstraction.
Logic, or mathematical rigor, is one of the most notable features of mathematics. In short, it means that any research in this science is susceptible to proof checking that may with 100% confidence state whether it is right or wrong. Although this idea already existed in antiquity, it was a kind of unattainable ideal until the beginning of the 20th century, when calculating machines made it possible to actually proof check almost any mathematical construction.
Despite its abstract nature, throughout history mathematics has been used in application to objective reality in many spheres of human activity, from architecture to warfare, thus giving birth to the field of applied mathematics of contemporary times. Although in the minds of many mathematicians there remains something far-away and unattainable, even the pure mathematical studies often lead to discoveries that find way into our everyday life. In some curious sense mathematics, while studying the subject matter that does not correspond to any physical reality, serves as a medium between almost all the other sciences, for many problems that arise from them find the realization of their solution in mathematics and nowhere else.