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Mathematics: Its Content, Methods and Meaning (3 Volumes in One) (Dover Books on Mathematics)

4.6 4.6 out of 5 stars 235 ratings
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". . . Nothing less than a major contribution to the scientific culture of this world." — The New York Times Book Review
This major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and including material on both elementary and advanced levels, encompasses 20 prime subject areas in mathematics in terms of their simple origins and their subsequent sophisticated developement. As Professor Morris Kline of New York University noted, "This unique work presents the amazing panorama of mathematics proper. It is the best answer in print to what mathematics contains both on the elementary and advanced levels."
Beginning with an overview and analysis of mathematics, the first of three major divisions of the book progresses to an exploration of analytic geometry, algebra, and ordinary differential equations. The second part introduces partial differential equations, along with theories of curves and surfaces, the calculus of variations, and functions of a complex variable. It furthur examines prime numbers, the theory of probability, approximations, and the role of computers in mathematics. The theory of functions of a real variable opens the final section, followed by discussions of linear algebra and nonEuclidian geometry, topology, functional analysis, and groups and other algebraic systems.
Thorough, coherent explanations of each topic are further augumented by numerous illustrative figures, and every chapter concludes with a suggested reading list. Formerly issued as a three-volume set, this mathematical masterpiece is now available in a convenient and modestly priced one-volume edition, perfect for study or reference.
"This is a masterful English translation of a stupendous and formidable mathematical masterpiece . . ." — Social Science
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  1. ISBN-10
    0486409163
  2. ISBN-13
    978-0486409160
  3. Publisher
    Dover Publications
  4. Publication date
    July 7, 1999
  5. Part of series
    Dover Books on Mathematics
  6. Language
    English
  7. Dimensions
    5.51 x 2.17 x 8.46 inches
  8. Print length
    1120 pages
  9. See all details
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Mathematics, Content, Methods, Meaning
Notes About The Text
  • Formerly issued as a three-volume set, now 1 volume; Over 1100 pages!
  • Reprint of the three-volume work by The M.I.T. Press, Cambridge, MA, 1963.

This major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and including material on both elementary and advanced levels, encompasses 20 prime subject areas in mathematics in terms of their simple origins and their subsequent sophisticated development.

Comments From The Esteemed Professor Morris Kline of New York University

"This unique work presents the amazing panorama of mathematics proper. It is the best answer in print to what mathematics contains both on the elementary and advanced levels."

Review From "The Mathematics Teacher"

"An excellent reference set for bright high school students and beginning college students . . . also of value to their teachers for lucid discussions and many good elementary examples in both familiar and unfamiliar branches. The intelligentsia of laymen who care to tackle more than today's popular magazine articles on mathematics will find many rewarding introductions to subjects of current interest."

Editorial Reviews

About the Author

The Russian Equation
Representative of the tremendous impact which Russian mathematicians have had on the Dover list since the Sputnik era is this outstanding book edited by A. D. Aleksandrov and others.

Critical Acclaim for Mathematics: Its Content, Methods and Meaning:
"In effect, these volumes present a do-it-yourself course for the person who would like to know what the chief fields of modern mathematics are all about but who does not aspire to be a professional mathematician or a professional user of mathematics. The coverage is extremely wide, including such important areas as linear algebra, group theory, functional analysis, ordinary and partial differential equations, the theory of functions of real and complex variables, and related subjects. . . . What makes these volumes so readable as compared with usual mathematics textbooks is the emphasis here upon basic concepts and results rather than upon the intricate and wearying proofs that make such demands in conventional textbooks and courses. There are proofs in these volumes, but usually they are presented only for the most important results, and even then to emphasize key areas and to illustrate the kind of methodology employed. . . . It is hard to imagine that any intelligent American with a curious mind and some good recollection of his high school and college mathematics would not find many entrancing discoveries in the intellectual gold mine that is this work." — The New York Times Book Review

"An excellent reference set for bright high school students and beginning college students . . . also of value to their teachers for lucid discussions and many good elementary examples in both familiar and unfamiliar branches. The intelligentsia of laymen who care to tackle more than today's popular magazine articles on mathematics will find many rewarding introductions to subjects of current interest." — The Mathematics Teacher

"Whether a physicist wishes to know what a Lie algebra is or 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." — Science

Product details

  • Publisher ‏ : ‎ Dover Publications (July 7, 1999)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 1120 pages
  • ISBN-10 ‏ : ‎ 0486409163
  • ISBN-13 ‏ : ‎ 978-0486409160
  • Item Weight ‏ : ‎ 2.71 pounds
  • Dimensions ‏ : ‎ 5.51 x 2.17 x 8.46 inches
  • Customer Reviews:
    4.6 4.6 out of 5 stars 235 ratings

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A. N. Kolmogorov

A. N. Kolmogorov

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4.6 out of 5 stars
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235 global ratings
Including Material On Both Elementary And Advanced Levels
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Including Material On Both Elementary And Advanced Levels
"This major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and INCLUDING MATERIAL ON BOTH ELEMENTARY AND ADVANCED LEVELS,....."[from the book of the back cover]
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Top reviews from the United States

Weston Buckhorn
5.0 out of 5 stars Companion Le Deaux
Reviewed in the United States on June 15, 2017
Verified Purchase
I think a table of contents would be useful. I use this book as a supplement to the "Princeton Companion to Mathematics", a companion companion as it were, um . . . companion squared? If we differentiate (companion)(companion) we get 2companion or "Companion Le Deaux".
VOLUME ONE, PART 1 CHAPTER I A GENERAL VIEW OF MATHEMATICS 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 II ANALYSIS M. A. Lavrent’ev and S. M. Nikol’ski?
§ 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 III ANALYTIC GEOMETRY B. N. Delone
§ 1. Introduction
§ 2. Descartes’ Two Fundamental Concepts
§ 3. Elementary Problems
§ 4. Discussion of Curves Represented by First-and 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 Conclusion
Suggested Reading

CHAPTER IV ALGEBRA: THEORY OF ALGEBRAIC EQUATIONS B. N. Delone
§ 1. Introduction
§ 2. Algebraic Solution of an Equation
§ 3. The Fundamental Theorem of Algebra
§ 4. Investigation of the Distribution of the Roots of a Polynomial on the Complex Plane
§ 5. Approximate Calculation of Roots
Suggested Reading

CHAPTER V ORDINARY DIFFERENTIAL EQUATIONS I. G. Petrovski?
§ 1. Introduction
§ 2. Linear Differential Equations with Constant Coefficients
§ 3. Some General Remarks on the Formation and Solution of Differential Equations
§ 4. Geometric Interpretation of the Problem of Integrating Differential Equations; Generalization of the Problem
§ 5. Existence and Uniqueness of the Solution of a Differential Equation; Approximate Solution of Equations
§ 6. Singular Points
§ 7. Qualitative Theory of Ordinary Differential Equations
Suggested Reading

VOLUME TWO, PART 3 CHAPTER VI PARTIAL DIFFERENTIAL EQUATIONS S. L. Sobolev and O. A. Ladyzenskaja
§ 1. Introduction
§ 2. The Simplest Equations of Mathematical Physics
§ 3. Initial-Value and Boundary-Value Problems; Uniqueness of a Solution
§ 4. The Propagation of Waves
§ 5. Methods of Constructing Solutions
§ 6. Generalized Solutions
Suggested Reading

CHAPTER VII CURVES AND SURFACES A. D. Aleksandrov
§ 1. Topics and Methods in the Theory
§ 3. Basic Concepts in the Theory of Surfaces
§ 4. Intrinsic Geometry and Deformation of Surfaces
§ 5. New Developments in the Theory of Curves and Surfaces
Suggested Reading

CHAPTER VIII THE CALCULUS OF VARIATIONS V. I. Krylov
§ 1. Introduction
§ 2. The Differential Equations of the Calculus of Variations
§ 3. Methods of Approximate Solution of Problems in the Calculus of Variations
Suggested Reading

CHAPTER IX FUNCTIONS OF A COMPLEX VARIABLE M. V. Keldyš
§ 1. Complex Numbers and Functions of a Complex Variable
§ 2. The Connection Between Functions of a Complex Variable and the Problems of Mathematical Physics
§ 3. The Connection of Functions of a Complex Variable with Geometry
§ 4. The Line Integral; Cauchy’s Formula and Its Corollaries
§ 5. Uniqueness Properties and Analytic Continuation
§ 6. Conclusion
Suggested Reading

PART 4 CHAPTER X PRIME NUMBERS K. K. Mardzanisvili and A. B. Postnikov
§ 1. The Study of the Theory of Numbers
§ 2. The Investigation of Problems Concerning Prime Numbers
§ 3. ?ebyšev’s Method
§ 4. Vinogradov’s Method
§ 5. Decomposition of Integers into the Sum of Two Squares; Complex Integers
Suggested Reading

CHAPTER XI THE THEORY OF PROBABILITY A. N. Kolmogorov
§ 1. The Laws of Probability
§ 2. The Axioms and Basic Formulas of the Elementary Theory of Probability
§ 3. The Law of Large Numbers and Limit Theorems
§ 4. Further Remarks on the Basic Concepts of the Theory of Probability
§ 5. Deterministic and Random Processes
§ 6. Random Processes of Markov Type
Suggested Reading

CHAPTER XII APPROXIMATIONS OF FUNCTIONS S. M. Nikol? ski?
§ 1. Introduction
§ 2. Interpolation Polynomials
§ 3. Approximation of Definite Integrals
§ 4. The ?ebyšev(Chebyshev) Concept of Best Uniform Approximation
§ 5. The ?ebyšev(Chebyshev) Polynomials Deviating Least from Zero
§ 6. The Theorem of Weierstrass; the Best Approximation to a Function as Related to Its Properties of Differentiability
§ 7. Fourier Series
§ 8. Approximation in the Sense of the Mean Square
Suggested Reading

CHAPTER XIII APPROXIMATION METHODS AND COMPUTING TECHNIQUES V. I. Krylov
§ 1. Approximation and Numerical Methods
§ 2. The Simplest Auxiliary Means of Computation
Suggested Reading

CHAPTER XIV ELECTRONIC COMPUTING MACHINES S. A. Lebedev and L. V. Kantorovi?
§ 1. Purposes and Basic Principles of the Operation of Electronic Computers
§ 2. Programming and Coding for High-Speed Electronic Machines
§ 3. Technical Principles of the Various Units of a High-Speed Computing Machine
§ 4. Prospects for the Development and Use of Electronic Computing Machines
Suggested Reading

VOLUME THREE, PART 5 CHAPTER XV THEORY OF FUNCTIONS OF A REAL VARIABLE S. B. Ste?kin
§ 1. Introduction
§ 2. Sets
§ 3. Real Numbers
§ 4. Point Sets
§ 5. Measure of Sets
§ 6. The Lebesgue Integral
Suggested Reading

CHAPTER XVI LINEAR ALGEBRA D. K. Faddeev
§ 1. The Scope of Linear Algebra and Its Apparatus
§ 2. Linear Spaces
§ 3. Systems of Linear Equations
§ 4. Linear Transformations
§ 5. Quadratic Forms
§ 6. Functions of Matrices and Some of Their Applications
Suggested Reading

CHAPTER XVII NON-EUCLIDEAN GEOMETRY A. D. Aleksandrov
§ 1. History of Euclid’s Postulate
§ 2. The Solution of Loba?evski?
§ 3. Loba?evski? Geometry
§ 4. The Real Meaning of Loba?evski? Geometry
§ 5. The Axioms of Geometry; Their Verification in the Present Case
§ 6. Separation of Independent Geometric Theories from Euclidean Geometry
§ 7. Many-Dimensional Spaces
§ 8. Generalization of the Scope of Geometry
§ 9. Riemannian Geometry
§ 10. Abstract Geometry and the Real Space
Suggested Reading

PART 6 CHAPTER XVIII TOPOLOGY P. S. Aleksandrov
§ 1. The Object of Topology
§ 2. Surfaces
§ 3. Manifolds
§ 4. The Combinatorial Method
§ 5. Vector Fields
§ 6. The Development of Topology
§ 7. Metric and Topological Spaces
Suggested Reading

CHAPTER XIX FUNCTIONAL ANALYSIS I. M. Gel? fand
§ 1. n-Dimensional Space
§ 2. Hilbert Space (Infinite-Dimensional Space)
§ 3. Expansion by Orthogonal Systems of Functions
§ 4. Integral Equations
§ 5. Linear Operators and Further Developments of Functional Analysis
Suggested Reading

CHAPTER XX GROUPS AND OTHER ALGEBRAIC SYSTEMS A. I. Mal? cev
§ 1. Introduction
§ 2. Symmetry and Transformations
§ 3. Groups of Transformations
§ 4. Fedorov Groups (Crystallographic Groups)
§ 5. Galois Groups
§ 6. Fundamental Concepts of the General Theory of Groups
§ 7. Continuous Groups
§ 8. Fundamental Groups
§ 9. Representations and Characters of Groups
§ 10. The General Theory of Groups
§ 11. Hypercomplex Numbers
§ 12. Associative Algebras
§ 13. Lie Algebras
§ 14. Rings
§ 15. Lattices
§ 16. Other Algebraic Systems
Suggested Reading

INDEX
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Rodion Melnikov
5.0 out of 5 stars Priceless overview of Mathematics, written by some of the best mathematicians of XX century
Reviewed in the United States on January 29, 2024
Verified Purchase
“If you can't explain it to a six year old, you don't understand it yourself.”
In this book, several of the best mathematicians of XX century explain their areas of research to high school graduates. That includes, for example, the genius Andrey Kolmogorov, who wrote the chapter on probability.

For me, this book is a comprehensive overview of undergraduate math that I took (and parts that I didn't) in extremely short, accessible, yet rigorous form. The chapters include great real-life examples, history moments, critical definitions and proofs. Unlike university courses, which are pressured to include as much of important material, this book is written to be interesting and to explain key concepts - something that course textbooks often badly miss.

Especially mind-blowing chapter is the one about analytic geometry. I never studied and never even seen anywhere a dedicated course on analytic geometry, its material was spread over various analysis, algebra, and differential equation courses, and never represented as a single consistent picture. However, from the chapter written by the genius Boris Delaunay it comes out as a fundamental math discipline, clear, intuitive and powerful, that explains many things done in other disciplines, which typically seem esoteric when taken out of geometric context.

Besides super-strong math content, this book is also an intriguing historical document. Though the translator removed most of the unrelated communist propaganda content that these world-class scientists had to include to make the book printable in the USSR, the chapters still include some interesting materialist explanations of math principles. I think they did honestly believe in it in some ways, so most of the explanations make sense, and makes me think that maybe the view of mathematics as a pure abstract science, and not part of natural sciences, is incomplete, if not outright wrong.

I can strongly recommend this book to young people considering choosing math or math-related major. I give the same recommendation to anyone who, like me, did study math but feels like misses something important.
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Amazon Customer
5.0 out of 5 stars Great compendium
Reviewed in the United States on January 3, 2024
Verified Purchase
This book is a great collection of mathematical topics. It is NOT a text of sequencial exposure like Apostol or Thomas, etc. Scan the Table of Contents to see if the topics interest you. Calculus, Differential Equations, Linear Algebra, and Discrete Math all make appearances. I'm not a math professor, I just like mathematics so I'm a sucker for a nice book like this. PS: The index is a full 20 pages.
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Rubisco
4.0 out of 5 stars Quick reference, not much more
Reviewed in the United States on March 20, 2023
Verified Purchase
This is a good book, in the sense that it is understandable and it does cover a lot.

The fact that it covers a lot is both the best and the worst part, really. It covers so much, that, even at it's pretty hefty size, the coverage of each topic is quite shallow.

Still, for the prize, I can't bring myself to give it less than 4 starts.
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Ana P.
5.0 out of 5 stars Letra pequena, livro bom.
Reviewed in Brazil on March 19, 2021
Verified Purchase
Gostei do livro, porém achei a letra muito pequena.
2 people found this helpful
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Ravi
5.0 out of 5 stars This is one of the best math books I have ever read
Reviewed in Canada on February 4, 2016
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This is one of the best math books I have ever read. I did not expect it to be so good.
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Oldtimer
5.0 out of 5 stars A must read for anyone who loves mathematics - beware, its not a light read though
Reviewed in India on July 24, 2016
Verified Purchase
This is quite simply the best exposition of mathematical beauty and concepts underlying various mathematical domains

If you love mathematics, ignore the price - you must read this.
Even if you are going to take a lifetime

Not for the lighthearted or the casual reader. Several topics require some familiarity with concepts that a person encounters at graduate and postgraduate levels
Nevertheless this book is not strewn with intimidating equations or proofs. Its perhaps the best that can be achieved in explaining the conceptual essence in the english language.
Immensely mind-opening, also gives a terrific historical perspective and helps connect the interconnectedness and the evolution of mathematics as a discipline itself

A must read for anyone who loves mathematics - beware, its not a light read though
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emew
5.0 out of 5 stars A Unique Classic of Mathematical Literature
Reviewed in Germany on June 9, 2013
Verified Purchase
Imagine a book on "mathematics", wherein the chapter on probability is written by Andrei Kolmogorov, the chapter on topology by Pavel Aleksandrov, the chapter on functional analysis by Israel Gelfand, and the chapter on partial differential equations by Sergei Sobolev and Olga Ladyzenskaja, to name just four of altogether 20 chapters, and just five of altogether 18 famous mathematicians, who are the authors of this work. Although written in an informal style, the book is worth being read not only by serious students and mathematically interested laypersons, but also by educators at the university level. E.g., everyone who lectures on probability and statistics will benefit from reading Kolmogorov's chapter on probability theory.

Every mathematician as well as every mathematically interested non-mathematician will like this masterpiece of mathematical exposition. Moreover, this Dover edition (three volumes bound as one, more than 1100 pages) is really worth its price.
6 people found this helpful
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Philip Jackson
5.0 out of 5 stars Mathematics Its Content, Methods and Meaning
Reviewed in the United Kingdom on March 7, 2013
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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.
6 people found this helpful
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