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What Is Mathematics? An Elementary Approach to Ideas and Methods [Paperback]

Richard Courant , Herbert Robbins , Ian Stewart
4.6 out of 5 stars  See all reviews (47 customer reviews)

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Book Description

July 18, 1996 0195105192 978-0195105193 2
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

Amazon.com Review

A 1996 revision of a timeless classic originally published in 1941. Highly recommended for any serious student, teacher or scholar of mathematics.


*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

Product Details

  • Paperback: 592 pages
  • Publisher: Oxford University Press; 2 edition (July 18, 1996)
  • Language: English
  • ISBN-10: 0195105192
  • ISBN-13: 978-0195105193
  • Product Dimensions: 9 x 6 x 1.1 inches
  • Shipping Weight: 1.7 pounds (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (47 customer reviews)
  • Amazon Best Sellers Rank: #39,316 in Books (See Top 100 in Books)

Customer Reviews

4.6 out of 5 stars
4.6 out of 5 stars
Most Helpful Customer Reviews
198 of 205 people found the following review helpful
5.0 out of 5 stars Excellent Book. Belongs on Your Bookshelf. August 3, 2003
Courant's 500-page text is not entirely suitable for the layman. Its target audience includes those who enjoy reading and studying mathematics and have a good background through precalculus or higher. "What is Mathematics?" is a mathematics book, not a book about mathematics.

"What is Mathematics?" is not a new book. It was first published by Oxford University Press in 1941 with later editions in 1943, 1945, and 1947. Good quality soft cover copies are still in print as Oxford Paperbacks.

The authors indicate that it is no means necessary to "plow through it page by page, chapter by chapter". I fully agree. I have skipped around, jumping to chapters of particular interest, but I have now read nearly every chapter.

I initially skipped to page 165 and delved directly into projective geometry (chapter IV), proceeded to topology (chapter V), and then jumped backwards to the beginning to explore the theory of numbers. After moving to geometry, I finally returned to the later chapters on functions and limits, maxima and minima, and the calculus.

Courant engages the reader in discussions on mathematical concepts rather than focusing on applications and problem solving. "What is Mathematics?" is a great textbook for students that have completed a year or more of calculus and wish to pull all of their mathematical learning together before moving on to more advanced studies. I suspect that it would even be welcomed by students that have completed an undergraduate degree in mathematics.

I cannot resist quoting Albert Einstein's comment on What is Mathematics? - "A lucid representation of the fundamental concepts and methods of the whole field of mathematics...Easily understandable."

Richard Courant was a highly respected mathematician.
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92 of 93 people found the following review helpful
By twit
This book genuinely has more mathematical content, for around $15-$25, than most, maybe all, "bridge" texts for college math majors, costing 5 or 10 times as much.

This book was written by a master, for an intelligent person knowing only 1950's style high school mathematics (some trig, algebra, and geometry).

When I fiorst tried to read it as a youngster however I was not used to books that required actually thinking about each statement, before proceeding to the next. Hence I could not read it at the pace I thought normal.

So this is not a breezy read, but is an outstanding one. It has literally no competitor to my knowledge at the present time, in quantity of material, quality of material, and quality of exposition.

Even experts may learn something here about the most familiar topics. E.g. in presenting the proof of the well known fact that all integers greater than one have unique prime factorizations, the authors show how a clever use of induction avoids developing the characterization of a gcd, which usually precedes this theorem. I had never seen that before.

If you are looking for a miracle book that treats the reader like a baby, and still covers calculus, this is not it. But if you have the prerequisites of a good high school course of elementary math, and are willing to spend time on the arguments, there is no better book for beginners and intelligent laypersons.
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70 of 72 people found the following review helpful
5.0 out of 5 stars Inspiring October 23, 1999
By A Customer
Although I was always good in math in high school, I never really appreciated it. One summer I found this book in a dusty little corner of a bookshelf and I started reading it. I still remember how for the first time, I was inspired by the subject while reading this book. I couldn't stop reading it, until I finished it. At the time, I didn't really know Calculus or any advanced subject and I had never read any math books other than the high school textbooks. This book literally changed my life. I might have forgotten who my first love was, but I remember very well this book after 25 years!
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58 of 61 people found the following review helpful
5.0 out of 5 stars Timeless. April 15, 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 presented with an eye to applications. The book is a classic and a masterpiece. The co-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. The book has been used by generations of readers, and it still points to the future.
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33 of 33 people found the following review helpful
5.0 out of 5 stars A Masterpiece June 23, 2000
If you start to read "What is Mathematics?" in order to find a direct answer to the title's issue, forget it! I would like to adapt a piece of "My Brain is Open", by Bruce Schechter, in the following way: "Asking a mathematician to explain exactly what is mathematics is a little like asking a poet what a poem is, or a musician what jazz is. Asked this last question, Louis Armstrong replied, `Man, if you gotta ask, you'll never know.'" On the other hand, if you start to read just to go deeper and deeper in the beautiful, and sometimes magic, structure of Math than I say: Go ahead! Because this book is a perennial source of pleasure. Of course it demands a lot of work to solve some of its problems (at least for me!), but as Courant says, you cannot learn music only by listening! I have reproduced almost all the calculations of this book and I know that it demands a lot of effort, but it is one of the few books I know where each small piece of calculation has its own reward! This book is my definition of perfect guide to Math style! Try it!
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Most Recent Customer Reviews
4.0 out of 5 stars Very good book
Is a very good book, some things could be better explained but is pretty good, lacks in spanish translation :(
Published 24 days ago by Alexev
5.0 out of 5 stars THE book for the budding mathematician.
This classic gets better and better.
It has no rival as a broad introduction to
the many fields in mathematics.
Published 1 month ago by Sally100
5.0 out of 5 stars Excellent book¡¡¡
It explains in a simple way many interesting subjects of mathematics of interest for several fields. It is a basic text book.
Published 1 month ago by F. C. L.
5.0 out of 5 stars Requires Stamina To Complete, But Worth It
My first attempt to read this book was during my undergraduate studies. Twenty years later, I am now discarding my college textbooks and "What Is Mathematics" has... Read more
Published 3 months ago by G. S. Cole
4.0 out of 5 stars What is Mathematics? An Elementary Approach to Ideas and Methods
If you love mathematics, you will love this book. It is simple to read and gives hundreds of clear examples on solving mathematical problems and great insights into the history of... Read more
Published 11 months ago by Angel Ortiz
5.0 out of 5 stars Great book for getting back into (the fun of) math
Although the book's layout looks a little dated, the contents are not: it is just the right mix of history, proofs, method and explanation that is missing between simple school... Read more
Published 13 months ago by Maurits Westerik
5.0 out of 5 stars The CLASSIC
Courant is crystal-clear in covering an astoundingly wide spectrum of subjects with an unusual didactic touch. Read more
Published 17 months ago by André Gargoura
1.0 out of 5 stars Does not address or answer the question.
My wife who had a degree in mathematics and eventually went on to get a PhD in communications from Columbia, took an entire year on her own to work through just the first chapter. Read more
Published on May 21, 2012 by George Frank
1.0 out of 5 stars Its a mess!
The formulas in this book have a lot of errors, missing signs and characters makes them impossible to understand. Read more
Published on March 5, 2012 by Svein Rćstad
5.0 out of 5 stars A work of art
I can't do this book justice, as many reviewers before me have. As an undergrad who's majoring in math, this book helps to illuminate so many fields of mathematics that I had not... Read more
Published on June 26, 2011 by nmgamer42
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