 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 electrodynamics, the theory of magnetism, and thermodynamics. Very important branches of the mechanics of continuous media were developed extensively; only the hydrodynamics of an incompressible ideal fluid had been created as early as the 18th century by D. Bernoulli, Euler, D’Alembert, and Lagrange. The mathematical requirements of technology also grew rapidly. At the start of the 19th century these included problems of thermodynamics of steam engines, technical mechanics, and ballistics. The theory of partial differential equations and especially the potential theory were developed more intensively as the fundamental apparatus for the new branches of mechanics and mathematical physics. Most of the leading analysts of the first half of the century—Gauss, J. Fourier, S. Poisson, Cauchy, P. Dirichlet, G. Green, and M. V. Ostrogradskii—worked in this direction. Ostrogradskii laid the foundations of the calculus of variations for functions of several variables. Vector analysis arose as a result of investigations of the equations of mathematical physics by G. Stokes and other British mathematicians.
Despite the mechanistic conviction of the possibility of describing all natural phenomena with differential equations that prevailed in natural science at the beginning of the 19th century, probability theory was further developed under the pressure of practical requirements. Laplace and Poisson created a powerful new analytic apparatus for this purpose. P. L. Chebyshev gave a rigorous justification of the elements of probability theory and proved his celebrated theorem (1867) combining all previously known forms of the law of large numbers in one general formulation.
As was indicated above, in addition to working on the new problems posed by natural science and technology, mathematicians focused attention on problems of the rigorous substantiation of analysis (Cauchy, 1821, 1823) from the very start of the 19th century. Lobachevskii (1834) and, later, Dirichlet (1837) clearly formulated the definition of function as a wholly