What Is Mathematics?: An Elementary Approach to Ideas and Methods / Edition 2

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For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.

Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.

Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.

Formal mathematics is like spelling and grammar—a matter of the correct application of local rules. Meaningful mathematics is like journalism—it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature—it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature—it opens a window onto the world of mathematics for anyone interested to view.

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Editorial Reviews

From the Publisher
*Praise for the previous edition:

"Without doubt, the work will have great influence. It should be in the hands of everyone, professional or otherwise, who is interested in scientific thinking."—The New York Times

"Should prove a source of great pleasure and satisfaction."—Journal of Applied Physics

"Succeeds brilliantly in conveying the intellectual excitement of mathematical inquiry and in communicating the essential ideas and methods."Journal of Philosophy

"It is a work of high perfection, whether judged by aesthetic, pedagogical or scientific standards. It is astonishing to what extent What is Mathematics? has succeeded in making clear by means of the simplest examples all the fundamental ideas and methods which we mathematicians consider the life blood of our science."—Herman Weyl

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Product Details

  • ISBN-13: 9780195105193
  • Publisher: Oxford University Press, USA
  • Publication date: 7/28/1996
  • Edition description: REV
  • Edition number: 2
  • Pages: 592
  • Sales rank: 385,728
  • Product dimensions: 9.00 (w) x 6.06 (h) x 1.05 (d)

Meet the Author

The late Richard Courant, headed the Department of Mathematicas at New York University and was Director of the Institute of Mathematical Sciences—which has subsequently renamed the Courant Institute of Mathematical Sciences. His book Mathematical Physics is familiar to every physicist, and his book Differential and Integral Calculus is acknowledged to be one of the best presentations of the subject written in modern times. Herbert Robbins is New Jersey Professor of Mathematical Statistics at Rutgers University. Ian Stewart is Professor of Mathematics at the University of Warwick, and author of Nature's Numbers and Does God Play Dice?. He also writes the "Mathematical Recreations" column in Scientific American.

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Table of Contents

Preface to Second Edition
Preface to Revised Editions
How to Use the Book
What is Mathematics?
Ch. I The Natural Numbers 1
Supplement to Chapter I. the Theory of Numbers 21
Ch. II The Number System of Mathematics 52
Supplement to Chapter II. The Algebra of Sets 108
Ch. III Geometrical Constructions. The Algebra of Number Fields 117
Ch. IV Projective Geometry. Axiomatics. Non-Euclidean Geometries 165
Ch. V Topology 235
Ch. VI Functions and Limits 272
Supplement to Chapter VI 322
Ch. VII Maxima and Minima 329
Ch. VIII The Calculus 398
Supplement to Chapter VIII 462
Ch. IX Recent Developments 487
Appendix: Supplementary Remarks, Problems, and Exercises 525
Suggestions for Further Reading 549
Suggestions for Additional Reading 553
Index 559
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Customer Reviews

Average Rating 4.5
( 8 )
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Sort by: Showing all of 8 Customer Reviews
  • Anonymous

    Posted January 15, 2003

    a superb book

    I love this book. The treatment is masterly and the topics are interesting. You can find things discussed here that will never be mentioned in standard treatments of these topics. Moreover they are discussed here in a self contained, if brief, style. The treatment omits some details, and proceeds quickly. As a young math major I felt embarrassed by how difficult it could be to read it, even about elementary topics. I was also surprized at finding things I did not know in a book intended for the lay public. As a mature mathematician, I have learned to read more slowly, and appreciate that many of the topics simply do not occur elsewhere at a similarly accessible level. If the book is hard for a true beginner, perhaps it is a good choice for a teacher of an introductory course, even an advanced introduction to the ideas of mathematics. For example, last night while preparing my "proofs" class, I wondered why in 2,000 years no one had given a proof of unique factorization of integers into primes, different from Euclid's original proof. As soon as I turned to the appropriate section in Courant-Robbins I found a more recent proof, clever in its avoidance of the concept of gcd, which is used in every other proof I had seen. I am now motivated to present this argument in my course, and to recommend this book. On another occasion I fielded a phone call with an interesting question about Appolonius' problem on circles tangent to a given set of circles. I struggled with an answer, then found a beautiful, succint discussion in Courant - Robbins. This is a work by a real master, both of the mathematics and of exposition. It is not easy, and not perfect, but I know of no substitute for it.

    2 out of 2 people found this review helpful.

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  • Anonymous

    Posted April 16, 2003


    Einstein writes...'Easily understandable.' And Herman Weyl,...'It is a work of high perfection.' It is both for beginners and for scholars. The first edition by Courant and Robbins, has been revised, with love and care, by Ian Stewart. Of the sciences, math stands out in the way some central ideas and tools are timeless. Key math ideas from our first mathematical experiences, perhaps early in life, often have more permanence this way. While the fads do change in math, there are some landmarks that remain, and which inspire generations. And they are as useful now as they were at their inception, the fundamentals of numbers, of geometry, of calculus and differential equations, and more. Much of it is presenterd with an eye to applications. The authors are ambitious (and remarkably sucessful)in trying to cover the essetials within the span of 500 plus pages. You find the facts, presented in clear and engaging prose, and with lots of illustrations.

    1 out of 1 people found this review helpful.

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  • Anonymous

    Posted July 29, 2001


    I give this book 5 stars because it is a classic. I believe, however, that it is too sketchy to be useful for the beginner as it is advertised. For chapter 1, for example, on number theory, I recommend Hardy's 'Introduction to the Theory of Numbers.' For the second chapter, on the number systems, I recommend a book like Birkhoff and MacLane's 'Modern Algebra.' It's difficult to write a survey of mathematics textbook without being sketchy and Courant isn't up to the task. In addition, the bibliography at the end of the book is fairly outdated, although the two books I mentioned above are included there. I also wish Courant would have provided more information on the evolution of mathematical concepts and ideas. This is something Kline does in his 'History of Mathematical Thought.' I find this information vital in answering the question 'what is mathematics?' If you really want to get a good idea of what mathematics is you should start with a general history of mathematics like Kline's book and quickly move on to Greek mathematics. Even a small understanding of Euclid's axiomatic method will help you understand modern day mathematics and why mathemticians do what they do the way they do it. Having said that, I plan on making more use of Courant's book later on in my mathematics career.

    1 out of 1 people found this review helpful.

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