Bibliography

S. N. Bernstein. Probability Theory. GIZ, 1927. Price 5 rubles 50 kopecks.

The appearance in print of this book, awaited with such impatience, is not only an event in the life of our higher educational institutions, but must also be noted as an important stage in the world’s scientific and educational literature on probability theory and its applications.

In its development, probability theory has lagged considerably behind other mathematical disciplines, and in essence only in the very most recent years has Europe begun to regard it as a science deserving the full interest and utmost attention of the mathematician. (In Russia such a view had become established already from the time of Chebyshev’s works.) Probability theory was regarded as a collection of entertaining, often amusing problems, which, to be sure, had great practical significance—something that was always acknowledged. Accordingly, the manuals that existed until now in Europe either had the character of “collections of mathematical entertainments” (the French courses, not excluding Poincaré’s brilliant book), or else were collections of practical instructions for workers in applied fields (German and English courses), and therefore naturally paid little attention to arousing and satisfying the mathematician’s interest: the absence of mathematical rigor and coherence, the inability to isolate and emphasize the mathematical idea of one problem or another—all this made them alien to the mathematician.

Among the Russian manuals that have existed, two should be noted: Markov’s course, theoretically very valuable, but pedagogically leaving much to be desired because of its heaviness, and, moreover, leaving the task of ideologically unifying probability theory and its various applications entirely unresolved; and Lakhtin’s course, written with great pedagogical tact and quite accessible to our students, but composed, as to its content, on the model of German courses, i.e. not designed to awaken mathematical interest and at every step leaving the critically minded reader in some perplexity.

Thus the compiler of the new course naturally faced a very difficult and responsible task: to combine in one manual theoretical rigor and depth, elementary character, liveliness and clarity of exposition, and at the same time to devote sufficient attention to applications, organically connecting them with the theoretical foundations.

It must be acknowledged that this complex task is brilliantly solved in the course written by Academician S. N. Bernstein—a course profoundly original, exceptionally valuable both scientifically and pedagogically, and able to combine depth of thought with simplicity of exposition in a way accessible only to first-rate thinkers. In this course, the mathematician and the practitioner will at last understand one another; they will understand that the same law of nature which, in its abstract form, excites and captivates mathematical thought, in its concrete manifestations constitutes the vital subject of study for the practitioner and applied worker; they will understand that they have studied and are studying one and the same thing.

The course begins with an axiomatic foundation of probability theory (which, so far as I know, has not yet existed in any textbook). Perhaps some mathematicians will find something to reproach in this axiomatics: it is not typical of contemporary mathematics, since it pursues no formal-logical aims. Its task is a substantive, concretely clear analysis, intelligible to every natural scientist, of the basic concepts and principles connected with the laws of chance—and this task has been carried out by the author brilliantly. In the very first chapter such concepts, fundamental for all applications, are introduced as the coefficients of regression and correlation; this connection between the formal theory and applied material so deeply and organically permeates the whole book that it is sometimes hard to say where one ends and the other begins; one can only state definitely that both sides gain considerably from this synthesis. The remarkable chapters on the law of large numbers and Laplace’s theorem are written in such a way that every mathematician will undoubtedly be carried away by them; and at the same time they contain everything, and only that, which directly serves applied aims.

From a pedagogical point of view the book leaves nothing to be desired; without any reservations it may be recommended to every student. Special mention should be made of the merits of the style: besides its general clarity and expressiveness, the author’s language is remarkable in that here the obsolete, awkward, and sometimes alien to the spirit of our language terms of probability theory used until now—such as “statochnost’,” “chastost’,” and the like—have been discarded.

S. N. Bernstein’s excellent book deserves the widest circulation.

A. Khinchin.