readme.txt

[什么是数学：对思想和方法的基本研究（增订版）]

ISBN: 0195105192
Title: What Is Mathematics?: An Elementary Approach to Ideas and Methods
Author: Richard Courant, Herbert Robbins and Ian Stewart
Publisher: Oxford University Press, USA
Publication Date: 1996-07-18

http://www.amazon.com/gp/product/0195105192/

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

Editorial Reviews

Albert Einstein
"A lucid representation of the fundamental concepts and methods of the whole field of mathematics...Easily understandable." (Editor's note: "hmmm...")


Review
"A lucid representation of the fundamental concepts and methods of the whole field of mathematics. It is an easily understandable introduction for the layman and helps to give the mathematical student a general view of the basic principles and methods."--Albert Einstein
"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
"A work of extraordinary perfection."--Mathematical Reviews
"It contains an excellent selection of material for students who have no desire to develop mathematical skills but who may be willing to look briefly into this field of intellectual activity....For the inquiring student who wishes to know what real mathematics is about, or for the trained engineer or physicist who has some interest in the justification of procedures he uses, it should prove a source of great pleasure and satisfaction."--Journal of Applied Physics
"This book is a work of art."--Marston Morse
"This is not a book in philosophy; but there are probably few philosophers who can not gain instruction and clarification from it. It 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
"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
"A great book."--Ludwig Otto, Paul Quinn College
"Can...be read with great profit by anyone desiring general mathematical literacy."--Mathematics Abstracts

Book Description
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 such as the Four-Color Theorem and Fermat's Last Theorem: open problems when Courant and Robbins wrote this masterpiece, but 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.

About the Author
The late Richard Courant, headed the Department of Mathematics 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 Professor of Mathematics at Columbia University. Ian Stewart is Professor of Mathematics at the University of Warwick.

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Customer reviews:

1 of 1 people found the following review helpful:
Unity and beauty, December 14, 2006
Reviewer:	Viktor Blasjo - See all my reviews
(REAL NAME)   
This is an excellent display of the beauty and unity of mathematics. Take geometry for example. Euclidean geometry is pretty but ruler-and-compass constructions have limitations. Their precise nature can only be understood by recasting geometry in terms of algebra through coordinate systems. This change of perspectives enables us to see clearly that it is impossible to trisect an angle or double a cube but that it is possible to construct a circle tangent to any three given circles (Apollonius's problem). Some fools think of this as a linear development ending with the triumph of algebra, but our authors know better, letting the pendulum swing back, through the beautiful topic of circle inversion, which can be carried out with compasses alone. This leads to a beautiful geometric solution of Apollonius's problem and construction of linkage devises that turn circular motion into rectilinear motion, desirable for engineering applications. Then there is of course a thorough discussion of projective geometry; this is a perfect topic for a book like this because of its great beauty and deep connections with many other areas of mathematics. Lines look like lines projectively, but circles don't have to look like circles, so projective geometry is in a way a "ruler only" geometry. Indeed, it adds to our discussion of constructions by revealing, as a simple consequence of the invariance of the cross-ratio, the theorem on the complete quadrilateral, which can be used, for example, to give a ruler-only construction of the continuation of a line beyond an obstacle. So the fact that we still have rulers in projective geometry gives a new fruitful perspective on old things. And the the fact that we don't have compasses since circles project to conic sections is also rewarding since it means that the theory of conic sections is unified by the projective viewpoint; in particular, this should imply that there are ruler-only constructions of conic sections and indeed we find a beautiful way of carrying out such constructions by projective pencils. Projective geometry shines even more as we study the projective disc model of hyperbolic geometry, but unfortunately our excursion into non-Euclidean geometry is very brief. The chapters on the calculus are also very nice and include for example a derivation of Leibniz' formula for pi (by series expansion of the arctangent) and Bernoulli's law-of-refraction solution of the brachistochrone problem, thus beautifully bringing together calculus, geometry, mechanics and optics, and the very last section of the book is yet another triumphant success for the unity of mathematics: chapter 1 was on elementary number theory and we now end with a heuristic proof of a result hinted at then, the prime number theorem (prime counting function is asymptotically equal to x/log(x)). First, log(n!) is asymptotically equal to n(log(n)) (proved by rectangular estimation of the integral of log(n!)), and log(n!) contains information about primes as follows. How many times does p divide n!=n(n-1)(n-2)...? It divides every p:th number once, every (p^2):th number once more, etc., so the answer is (n/p)+(n/p^2)+...=n/(p-1), thus, writing n! as its prime factorisation, we have n!=product of p^(n/(p-1)), and taking the logarithm we get log(n!)=sum of (n/(p-1))log(p). But log(n!) "=" n(log(n)) by the above, so log(n)=sum of log(p)/(p-1)=integral of prime density times log(x)/(x-1). Differentiating both sides and solving for prime density gives (x-1)/(x(log(x)), which has asymptotical antiderivative x/log(x). QED.





	

4 of 4 people found the following review helpful:
My favorite math book, October 14, 2006
Reviewer:	Gary Lyndaker (Gravois Mills, MO USA) - See all my reviews
(REAL NAME)   
I read the 1961 edition of the book (in 1965) and it introduced me to the beauty and challenge of mathematics. It is mainly targetted at those who are serious about understanding, at an introductory level, the broad scope of mathematics. It also presents the beauty of mathematics far beyond the arithmetic, basic algebra, and trigonometry that dominate high school math. This book was instrumental in giving me a lifelong love of mathematics.





	

0 of 3 people found the following review helpful:
math is important, August 25, 2006
Reviewer:	Jerome D. Gorman - See all my reviews
(REAL NAME)   
an excellent help in understanding and exploring and using
mathematics more effectively





	

21 of 28 people found the following review helpful:
The Invisible Second Author, September 1, 2005
Reviewer:	H. G. Sung (Houston, TX) - See all my reviews
(REAL NAME)   
The purpose of this review is to bring your attention to the second author of this timeless classic. Apparently most reviewers give all the credit to the first author, Richard Courant. Allow me to bring the second author, Herbert Robbins, to your attention. Google his name and you will find that Herbert Robbins is one of the most prolific and creative statisticians ever existed. Robbins studied mathematics at Harvard in the 1930s. At the time he worked with Courant on this book, he was a young rising star in mathematics/statistics. I have every reason to believe that Robbins has done more to this book than we give him credit for. We may never know the exact magnitude of Robbins's contribution to this book, but a complete ignorance of him is certainly unjust.






	

20 of 20 people found the following review helpful:
the best bargain in introductory math books in existence, April 20, 2005
Reviewer:	twit - See all my reviews
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.





	

21 of 22 people found the following review helpful:
Pretty good, December 17, 2003
Reviewer:	"jscollier" (Round Rock, Texas United States) - See all my reviews
For the most part I love this book. It is informative, and relatively simple to understand. This book is an "elementary approach to ideas and methods" for the whole field of mathmatics. In fact, this book is one of the reasons I changed my major to mathmatics.

However, there are two main problems with this book. First the quality of the print varies. Occasionally, whole sets of subscripts are blurred, which makes understanding the equation of the moment difficult, if not impossible.

Second, the order of steps for solving or understanding a problem are in an unexpected order, which is confusing. Often, I find that a difficult passage doesn't deal with difficult concepts, its just that the concepts are explained in an unusual way.

Aside from those problems, this is an extraordinary introduction to mathmatics.





	

13 of 26 people found the following review helpful:
Not an 'easy' read, August 21, 2003
Reviewer:	newton fisher "nerdly uncki" (riverside, california United States) - See all my reviews
(REAL NAME)   
This is a demanding book. One cannot read it listening to Bach. [a clash of complexities]. The construction of he book is `old style' [which is every seeming possible variation is mentioned] which has fallen into disfavor as confusing,

That written it is very complete and I really enjoyed many parts of this book.





	

77 of 79 people found the following review helpful:
Excellent Book. Belongs on Your Bookshelf., August 3, 2003
Reviewer:	Michael Wischmeyer (Houston, Texas) - See all my reviews
(TOP 1000 REVIEWER)    (REAL NAME)   
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. He taught in Germany and in Cambridge and was director of the Institute of Mathematical Sciences at New York University (now renamed the Courant Institute of Mathematical Sciences). Courant has authored other widely acclaimed mathematical texts including Methods of Mathematical Physics (co-authored with David Hilbert) and his popular Differential and Integral Calculus.