Randomization of Algebra and Algebraization of Probability

It is probably difficult to find areas of XIXth Century mathematics more remote from each other than algebra, the foundation of mathematics, and probability theory, a semi-applied area perceived from the time of its emergence as an almost experimental science. We note, in passing, that in the works of P. L. Chebyshev’s students, A. A. Markov and A. M. Lyapunov, many assertions of probability theory (for example, the central limit theorem) were proved with complete rigour in great generality. Nevertheless, it is no accident that one of the famous problems, proposed by D. Hilbert involved axiomatization of mechanics and axiomatization of probability theory: at that time one could not assume that these areas were fully mathematicized. In the first half of the XXth Century the works of A. N. Kolmogorov, S. N. Bernstein, von Mises et al. created the foundations of probability theory, which were unconditionally accepted by the mathematical community, and all doubts about whether or not this was mathematics were removed. However, probability theory has retained a certain isolation until now. It is difficult to explain this rationally. Certainly, a number of its methods are specific to that science and were difficult to understand even 20 years ago. For example, specialists on differential equations were for a long time unable to assimilate techniques of stochastic calculus, although results obtained by probabilistic methods in the theory of equations competed successfully with theorems obtained by classical methods.