Abstract
Abstract: Arithmetic is the study of numbers, their properties and relationships. Calculations related to the centralization of agriculture and counting had a practical impact on the formation of arithmetic. The first reliable information related to arithmetic was found in the historical monuments of Babylon and Ancient Egypt, dating back to the III-II millennia BC. In the Middle Ages, the sphere of trade and the conduct of approximate calculations were the main branches of the application of arithmetic. Initially, arithmetic developed in India and Muslim countries, and later began to spread in Western Europe. In modern times, marine astronomy, mechanics and complex commercial calculations posed new challenges and requirements, thereby contributing to new rounds in the development of astronomy. The article deals with the issues of comprehending the concept of number and its content in the framework of mathematical discourses of the 19th century. This issue is initially considered in a historical perspective, then the works of such mathematicians and philosophers as Richard Dedekind, Georg Cantor, Giuseppe Peano and the logic Gottlob Frege are analyzed. In the 1870s, the discourse on the foundations of arithmetic was updated. For the approval of mathematics as a consistent and correct science, the importance of determining the meaning of the concept of number, which is considered the basis of arithmetic, is noted. The article discusses questions about the essence and content of the concept of number. Key words: concepts of number, foundations of arithmetic, philosophy of mathematics, Dedekind, Frege, Peano, Cantor.