Table Of ContentMATHEMATICS
Its Contents, Methods and Meaning
THREE VOLUMES BOUND AS ONE
Edited by
A. D. Aleksandrov,
A. N. Kolmogorov,
M. A. Lavrent’ev
Translation edited by
S. H. Gould
DOVER PUBLICATIONS, INC.
Mineola, New York
Bibliographical Note
This Dover edition, first published in 1999, is an unabridged reprint in one
volume of second edition of Mathematics: Its Content, Methods and Meaning,
published in three volumes by The M.I.T. Press, Cambridge, MA, in 1969.
Library of Congress Cataloging-in-Publication Data
Matematika, ee soderzhanie metody i znachenie. English.
Mathematics, its content, methods, and meaning / edited by A.D.
Aleksandrov, A.N. Kolmogorov, M.A. Lavrent’ev; translation edited by
S.H. Gould.
p. cm.
Previously published: Cambridge, Mass., M.I.T. Press, 1964, cl963.
Includes index.
eISBN 13: 978-0-486-15787-0
1. Mathematics. I. Aleksandrov, A. D. (Aleksandr Danilovich),
1912-. II. Kolmogorov, A. N. (Andrei Nikolaevich), 1903-. III. Lavrent’ev,
Mikhail Alekseevich, 1900-. IV. Title.
QA36.M2913 1999
510–dc21
99-33023
CIP
Manufactured in the United States by Courier Corporation
40916308
www.doverpublications.com
PREFACE TO THE RUSSIAN EDITION
Mathematics, which originated in antiquity in the needs of daily life, has
developed into an immense system of widely varied disciplines. Like the other
sciences, it reflects the laws of the material world around us and serves as a
powerful instrument for our knowledge and mastery of nature. But the high level
of abstraction peculiar to mathematics means that its newer branches are
relatively inaccessible to nonspecialists. This abstract character of mathematics
gave birth even in antiquity to idealistic notions about its independence of the
material world.
In preparing the present volume, the authors have kept in mind the goal of
acquainting a sufficiently wide circle of the Soviet intelligentsia with the various
mathematical disciplines, their content and methods, the foundations on which
they are based, and the paths along which they have developed.
As a minimum of necessary mathematical knowledge on the part of the
reader, we have assumed only secondary-school mathematics, but the volumes
differ from one another with respect to the accessibility of the material contained
in them. Readers wishing to acquaint themselves for the first time with the
elements of higher mathematics may profitably read the first few chapters, but
for a complete understanding of the subsequent parts it will be necessary to have
made some study of corresponding textbooks. The book as a whole will be
understood in a fundamental way only by readers who already have some
acquaintance with the applications of mathematical analysis; that is to say, with
the differential and integral calculus. For such readers, namely teachers of
mathematics and instructors in engineering and the natural sciences, it will be
particularly important to read those chapters which introduce the newer branches
of mathematics.
Naturally it has not been possible, within the limits of one book, to exhaust all
the riches of even the most fundamental results of mathematical research; a
certain freedom in the choice of material has been inevitable here. But along
general lines, the present book will give an idea of the present state of
mathematics, its origins, and its probable future development. For this reason the
book is also intended to some extent for persons already acquainted with most of
the factual material in it. It may perhaps help to remove a certain narrowness of
outlook occasionally to be found in some of our younger mathematicians.
The separate chapters of the book are written by various authors, whose
names are given in the Contents. But as a whole the book is the result of
collaboration. Its general plan, the choice of material, the successive versions of
individual chapters, were all submitted to general discussion, and improvements
were made on the basis of a lively exchange of opinions. Mathematicians from
several cities in the Soviet Union were given an opportunity, in the form of
organized discussion, to make many valuable remarks concerning the original
version of the text. Their opinions and suggestions were taken into account by
the authors.
The authors of some of the chapters also took a direct share in preparing the
final version of other chapters: The introductory part of Chapter II was written
essentially by B. N. Delone, while D. K. Faddeev played an active role in the
preparation of Chapter IV and Chapter XX.
A share in the work was also taken by several persons other than the authors
of the individual chapters: §4 of Chapter XIV was written by L. V. Kantorovič,
§6 of Chapter VI by O.A. Ladyženskaja, §5 of Chapter 10 by A. G. Postnikov;
work was done on the text of Chapter V by O.A. Oleĭnik and on Chapter XI by
Ju. V. Prohorov.
Certain sections of Chapters I, II, VII, and XVII were written by V. A.
Zalgaller. The editing of the final text was done by V. A. Zalgaller and V. S.
Videnskiĭ with the cooperation of T. V. Rogozkinaja and A. P. Leonovaja.
The greater part of the illustrations were prepared by E. P. Sen’kin.
Moscow
1956 EDITORIAL BOARD
FOREWORD BY THE EDITOR OF THE
TRANSLATION
Mathematics, in view of its abstractness, offers greater difficulty to the
expositor than any other science. Yet its rapidly increasing role in modern life
creates both a need and a desire for good exposition.
In recent years many popular books about mathematics have appeared in the
English language, and some of them have enjoyed an immense sale. But for the
most part they have contained little serious mathematical instruction, and many
of them have neglected the twentieth century, the undisputed “golden age” of
mathematics. Although they are admirable in many other ways, they have not yet
undertaken the ultimate task of mathematical exposition, namely the large-scale
organization of modern mathematics in such a way that the reader is constantly
delighted by the obvious economizing of his own time and effort. Anyone who
reads through some of the chapters in the present book will realize how well this
task has been carried out by the Soviet authors, in the systematic collaboration
they have described in their preface.
Such a book, written for “a wide circle of the intelligentsia,” must also discuss
the general cultural importance of mathematics and its continuous development
from the earliest beginnings of history down to the present day. To form an
opinion of the book from this point of view the reader need only glance through
the first chapter in Part 1 and the introduction to certain other chapters; for
example, Analysis, or Analytic Geometry.
In translating the passages on the history and cultural significance of
mathematical ideas, the translators have naturally been aware of even greater
difficulties than are usually associated with the translation of scientific texts. As
organizer of the group, I express my profound gratitude to the other two
translators, Tamas Bartha and Kurt Hirsch, for their skillful cooperation.
The present translation, which was originally published by the American
Mathematical Society, will now enjoy a more general distribution in its new
format. In thus making the book more widely available the Society has been
influenced by various expressions of opinion from American mathematicians.
For example, “. . . the book will contribute materially to a better understanding
by the public of what mathematicians are up to . . . . It will be useful to many
mathematicians, physicists and chemists, as well as to laymen . . . . Whether a
physicist wishes to know what a Lie algebra is and how it is related to a Lie
group, or an undergraduate would like to begin the study of homology, or a
crystallographer is interested in Fedorov groups, or an engineer in probability, or
any scientist in computing machines, he will find here a connected, lucid
account.”
In its first edition this translation has been widely read by mathematicians and
students of mathematics. We now look forward to its wider usefulness in the
general English-speaking world.
August, 1964
S. H. GOULD
Editor of Translations
American Mathematical Society
Providence, Rhode Island
CONTENTS
VOLUME ONE, PART 1
CHAPTER A GENERAL VIEW OF MATHEMATICS
I
A. D. Aleksandrov
§1. The Characteristic Features of Mathematics
§2. Arithmetic
§3. Geometry
§4. Arithmetic and Geometry
§5. The Age of Elementary Mathematics
§6. Mathematics of Variable Magnitudes
§7. Contemporary Mathematics
Suggested Reading
CHAPTER ANALYSIS
II
M. A. Lavrent’ev and S. M. Nikol’skiĭ
§1. Introduction
§2. Function
§3. Limits
§4. Continuous Functions
§5. Derivative
§6. Rules for Differentiation
§7. Maximum and Minimum; Investigation of the Graphs of
Functions
§8. Increment and Differential of a Function
§9. Taylor’s Formula
§10. Integral
§11. Indefinite Integrals; the Technique of Integration
§12. Functions of Several Variables
§13. Generalizations of the Concept of Integral
§14. Series
Suggested Reading
PART 2
CHAPTER ANALYTIC GEOMETRY
III
B. N. Delone
§1. Introduction
§2. Descartes’ Two Fundamental Concepts
§3. Elementary Problems
§4. Discussion of Curves Represented by Firstand Second-
Degree Equations
§5. Descartes’ Method of Solving Thirdand Fourth-Degree
Algebraic Equations
§6. Newton’s General Theory of Diameters
§7. Ellipse, Hyperbola, and Parabola
§8. The Reduction of the General Second-Degree Equation to
Canonical Form
§9. The Representation of Forces, Velocities, and
Accelerations by Triples of Numbers; Theory of Vectors
§10. Analytic Geometry in Space; Equations of a Surface in
Space and Equations of a Curve
§11. A fine and Orthogonal Transformations
§12. Theory of Invariants
§13. Projective Geometry
§14. Lorentz Transformations
