ts and relations themselves. A theory is applicable to any system of objects connected by relations that satisfy its fundamental system of axioms. Accordingly, a theory is considered to be logically rigorous if it employs only those properties of the objects or of the relations that are mentioned in the axioms and if all additional objects or relations that are introduced in the course of the development of the theory are defined formally in terms of the axioms.
Mathematical logic clarifies another aspect of the structure of any mathematical theory. A system of axioms in the (set-theoretic) sense outlined above now merely imposes “outer” limits on the domain of applications of a given mathematical theory. It indicates the properties of the system of objects and relations under study but provides no clues concerning the logical means by which this mathematical theory may be developed. For example, the properties of the system of natural numbers can be determined up to the accuracy of isomorphism by using a very simple system of axioms. Nevertheless, it often proves to be difficult to solve problems the answer to which in principle is uniquely predetermined by the adoption of this system of axioms: number theory abounds in problems that were posed and very simply formulated long ago but that have not yet been solved. The question naturally arises whether this is so because the solution of a few simply formulated problems of number theory requires a very long chain of arguments composed of elementary links that are known and already have come into use, or because essentially new, not previously used, methods of logical derivation are required to solve certain problems of number theory.
Modern mathematical logic has provided a specific answer to this question: no single deductive theory can exhaust the diversity of the problems of number theory. More accurately, even within the theory of natural numbers it is possible to formulate a sequence of problems p1,p2, …, pn, … such that for any deductive theory there will be found among these problems one that is unsolvable within the limits of the given theory (K. Gödel). Here, “deductive theory” implies a theory that is developed from a finite number of axioms by constructing chains of arguments of arbitrary length consisting of links belonging to a finite number of elementary methods of logical deduction that are fixed for the given theory.
Thus it was found that the concept of a mathematical theory in the sense of a theory encompassed by a single system of set-theoretic axioms is substantially broader than the logical concept of a deductive theory: unlimited use of essentially new methods of logical arguments lying outside any finite set of standardized methods is inevitable even in the development of the arithmetic of natural numbers.
All results that can be obtained within the framework of a single deductive theory can also be obtained by calculations based on a fixed set of rules. If a rigorously defined prescription is given for the solution of some class of problems, then we speak of a mathematical algorithm. From the very creation of an adequately developed system of mathematical symbols, the problem of constructing sufficiently general and at the same time brief algorithms has occupied a major place in the history of mathematics. But only in the last decades have the general theory of algorithms and the theory of “algorithmic solvability” of mathematical problems been developed. The practical prospects for these theories are apparently great, particularly in connection with the current development of computing techniques that make it possible to replace complex mathematical algorithms with computer programs.
History of mathematics in the 19th century and early 20th century. FIRST HALF OF THE 19TH CENTURY. A considerable expansion of the domain of applications of mathematical analysis took place in the early 19th century. In physics before this, mainly mechanics and optics required a large mathematical apparatus; but now such an apparatus also became a requirement of elect