he various forms of spaces of arbitrary dimension. With this broad understanding of the terms “quantitative relations” and “spatial forms,” the definition of mathematics given at the beginning of the article is also applicable to the new, modern stage of the development of mathematics.
An essential innovation in 19th-century mathematical development was that problems related to the necessary expansion of the range of quantitative relations and spatial forms to be studied became the objects of conscious and active interest on the part of mathematicians. Whereas previously the introduction, for example, of negative and complex numbers and the precise formulation of the rules governing operations involving such numbers required much work and effort, mathematics now required the development of techniques for the planned creation of new geometric systems and new “algebras” with noncommutative or even nonassociative multiplication needed in one or another connection. For example, the creation of a new “algebra” with new rules of operation in order to analyze and synthesize some new type of relay-contact circuits is now a matter of everyday scientific technical practice and no longer generates particular surprise. It would be difficult to overrate the importance of the reorganization of the entire frame of mathematical thinking that had to take place during the 19th century for this change in attitude to set in. From this conceptual viewpoint, the discovery of Lobachevskii’s non-Euclidean geometry was the most significant discovery of the early 19th century. It was precisely on the basis of this geometry that belief in the inviolability of the axioms that had been elucidated by a millennium of mathematical development was overcome and the possibility of creating essentially new mathematical theories by selectively setting aside previously imposed restrictions that had no intrinsic logical necessity was understood; it was also perceived that with time such an abstract theory may find increasingly broad and quite specific applications.
The extraordinary expansion of the subject of mathematics in the 19th century resulted in increased attention to the problems of the “substantiation” of mathematics, that is, to a critical reevaluation of initial assumptions (axioms), to the construction of a rigorous system of definitions and proofs, and to a critical examination of the logical means used in these proofs. Works on the rigorous substantiation of various branches of mathematics rightly occupied a significant place in 19th- and 20th-century mathematics. The results obtained in the substantiation of the foundations of analysis (the theory of real numbers, the theory of limits, and the rigorous substantiation of all methods of the differential and integral calculus) are now given with greater or lesser completeness in most textbooks (even purely practical textbooks). However, there are still instances where the rigorous substantiation of a mathematical theory arising from practical requirements is delayed. As late as the turn of the 20th century this was the case with operational calculus, which found extremely broad applications in mechanics and electrical engineering. A logically irreproachable exposition of mathematical probability theory was given only after a long delay. And even now there is no rigorous substantiation of many mathematical methods that are used extensively in modern theoretical physics, where many valuable results are obtained by “illegal” mathematical procedures.
The standards of mathematical rigor currently observed by practicing mathematicians took shape only toward the end of the 19th century. These standards are based on the set-theoretic conception of the structure of any mathematical theory. From this point of view any mathematical theory deals with one or several sets of objects that are interconnected by certain relations. All formal properties of these objects and relations necessary for the development of the theory are stated in the form of axioms that do not touch upon the specific nature of the objec