In this work of over one thousand pages the authors have attempted a broad survey of the fields of mathematics as they were in the mid twentieth century. The book is highly readable and should be accessible to advanced undergraduates in the mathematical sciences. The main branches of mathematics including algebra, analysis, geometry, differential equations, complex analysis, number theory, approximation, linear algebra, non-euclidean geometry, measure theory, topology, functional analysis and group theory are all given chapters. I would recommend this work to anyone seeking an overview of mathematics which also contains some of the meat of the subject. Very informative.
In Romanian I found the old version of the book in separated 3 volumes. I've bought this English version because is printed in a new format (all 3 volumes printed in one volume) and because the quality of the printed paper is excellent. The content of the book doesn't need any other explanations, it is one of the few that covers most of the studied mathematics topics at undergraduate and graduate levels.
What communism did badly it did atrociously. What it did well, it did brilliantly. The achievements of Soviet mathematicians were part of that brilliance. Though originally published over half a century ago, this book is still fresh. Jointly written by the cream of the Steklov Mathematical Institute, this book attempts nothing less than a panoramic survey of the principal branches of mathematics.
The chapters and authors say it all:
1. A General View of Mathematics (A.D Aleksandrov) 2. Analysis (M. A. Lavrentev & S. M. Nikolski) 3. Analytic Geometry (B. N. Delone) 4. Algebra: The Theory of Algebraic Equations (B. N. Delone) 5. Ordinary Differential Equations (I. M. Petrovski) 6. Partial Differential Equations (S. L Sobolev & O. A. Ladyzhenskaya) 7. Curves and Surfaces (A. D. Alexandrov) 8. The Calculus of Variations (V. I. Krylov) 9. Functions of a Complex Variable (M. V. Keldys) 10. Prime Numbers (K. K. Mardzanisvili & A. B. Postnikov) 11. The Theory of Probability (A. N. Kolmogorov) 12. Approximation of Functions (S. M. Nikolski) 13. Approximation Methods and Computing Techniques (V. I. Krylov) 14. Electronic Computing Machines (S. A. Lebedev & L. V. Kantorovich) 15. Theory of Functions of Real Variable (S. B. Steckin) 16. Linear Algebra (D. K. Faddeev) 17. Non-Euclidean Geometry (A. D. Aleksandrov) 18. Topology (P. S. Aleksandrov) 19. Functional Analysis (I. M. Gelfand) 20. Groups and Other Algebraic Systems (A. I Malcev)
Lest the stature of the authors put you off, one can only say that they uphold the one of the better ideas from the Soviet era, namely that mathematics should not be taught in isolation from its history or its applications. The result is a supremely accessible survey that IMO should be read by any maths undergraduate if not before she begins her course, then at least during the first year.
To those who follow that piece of advice, I give another: Some of your university teachers may tell you that the book is too dated and that you should discard it in favour of one that adopts purely axiomatic approaches to each branch of mathematics. If so, you should lie in wait for them, ambush them and beat the idiots about the head with all 1000+ pages until they recant their heresy.
Dover has done Western mathematicians proud by making this wonderful book available!