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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"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." -- New York Times
"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." -- New York Times
About 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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Number is the basis of modern mathematics. Read the first page
Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
Number is the basis of modern mathematics. Read the first page
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Most Helpful Customer Reviews
7 of 7 people found the following review helpful
Format:Paperback
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. The authors are ambitious 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.
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. The authors are ambitious 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.
28 of 30 people found the following review helpful
By A Customer
Format:Paperback
A wide-ranging overview of pure mathematics, first published in the 1940s, now re-issued and brought up to date with an additional chapter by Ian Stewart, this book is at the level of a capable student at the higher end of high school mathematics (A-level in the UK) or the beginning of a mathematics degree.
The scope of this book is awesome, covering number theory, geometry, topology, calculus, and much more. The chapter on projective geometry is a real treat, as it explores a beautiful topic that has dropped out of the modern maths syllabus. The only noticeable omission is group theory, which gets only a passing mention.
The style is clear, although the pace is rapid, and the reader is expected to fill in some details. There is an emphasis throughout on rigour - where this is relaxed for the sake of brevity, this is clearly signalled.
An appendix of problems and exercises (without answers) encourages further exploration of each topic.
A true classic and an enhancement to any mathematician's bookshelf.
9 of 10 people found the following review helpful
By A Customer
Format:Paperback
So Einstein thought this book "easily understandable" ? Well, if you are a beginner at calculus you will not find it "easily understandable", for that would mean you didn't learn a single new thing! Calculus is perhaps the most profound and far-reaching discovery of the millenium, and is certainly not trivial. However, this magical book is the best possible introduction. It is written so that your perplexities will always be accompanied by so beautiful results or promises of results, that you will be more than ready to do the necessary efforts. These come, for instance, in the form of exercises and in the details of the demonstrations, which are all there. There is no cheating. Well, the book is not only about calculus. There are many previous chapters on theory of numbers, geometry, algebra, topology. But I think it culminates with calculus, and the preceding chapters serve as steps of a staircase leading to it. The new edition has the collaboratio! n of Ian Stewart, an inspired writer.
2 of 2 people found the following review helpful
By Enigma
Format:Paperback
I was searching for the perfect book in my first year of a 2 year BA Hons in Mathematics Education to really help me to extend my mathematical knowledge but in a readable format.
This was recommended to me by my Geometry lecturer as the bible to which he would always return and when I read it I could see why. Fantastically clear explanations which really get back to the roots of things like differentiation. I am sure that the fact the author used his (then)teenage son to help him in formulating understandable explanations shines through. This book might have first been written many decades ago, but it has certainly stood the test of time. I cant recommend it highly enough.
This was recommended to me by my Geometry lecturer as the bible to which he would always return and when I read it I could see why. Fantastically clear explanations which really get back to the roots of things like differentiation. I am sure that the fact the author used his (then)teenage son to help him in formulating understandable explanations shines through. This book might have first been written many decades ago, but it has certainly stood the test of time. I cant recommend it highly enough.
11 of 13 people found the following review helpful
By ksk
Format:Paperback
Forgiving my blasphemy above, this is an unsurpassably great maths book. This book is about beautiful maths beautifully written: it epitomises the romance and magnetism of mathematics. You will learn some fundamental maths supplemented with techniques as powerful as they are elegent. I'm a theorist, and I think all academics and amateurs should have this book - not just my (socially retarded) brothers, the mathematicians.
This book aquaints you with the pure fundamentals in the calculus, topology, limits and convergence, the number system........just buy this book, ok. If you're here, and thus considering buying this book, let me assure you that it is well worth it.
If you love mathematics, you will love this book; and if you love this book, you're in love with mathematics.
1 of 1 people found the following review helpful
By M. F. Cayley
Format:Paperback|Amazon Verified Purchase
The book covers a broad range of topics in (mainly pure) maths. It should be comprehensible to anyone who has done the equivalent of UK "A" Level maths. To get the most out of it, you need to be willing to concentrate hard - explanations are clear but sometimes at a breathless pace - and to tackle some of the exercises. The most technical parts are clearly flagged up as optional reading. For anyone wanting to go beyond school maths, or about to embark on a maths course at university, the book will be invaluable.
If I have a quibble, it is that the exercises and some parts of the book are printed in smallish type which my (ageing) eyes had difficulty coping with, especially when it came to subscripts and superscripts, some of which I could not decipher. In places the notation - for instance some symbols used in relation to sets - is outmoded, but this is not a significant problem.
If I have a quibble, it is that the exercises and some parts of the book are printed in smallish type which my (ageing) eyes had difficulty coping with, especially when it came to subscripts and superscripts, some of which I could not decipher. In places the notation - for instance some symbols used in relation to sets - is outmoded, but this is not a significant problem.
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4.6 out of 5 stars
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Most Recent Customer Reviews
4.0 out of 5 stars
A Good (if a little dense) Read
This book was great! It covered a huge range of mathematical ideas, and pushed past anything learned in school or college. The algebra is well laid out, and it is well-written. Read more
Published 1 month ago by Sam Symonds
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 5 months ago by André Gargoura
4.0 out of 5 stars
Not too basic, not too advanced
This is a book for scholars. If you want to learn some mathematics instead of just reading about it, then this is the book for you. Read more
Published 7 months ago by Paul P. Mealing
5.0 out of 5 stars
A lucid and exciting introduction indeed
What Courant and Robbins have managed to do is to write a book that not only introduces to some of the most fascinating aspects of mathematics, in a beautiful, challenging way, but... Read more
Published on 2 Nov 2009 by Jamie B.
4.0 out of 5 stars
Not for the beginner
Although well written this book isn't for someone who is looking to get into maths above what they learned at high school. Read more
Published on 8 Nov 2008 by M. Bradshaw
5.0 out of 5 stars
The best book on mathematics which I ever read!
This is a "must read" for every student of mathematics, engineering and the sciences. You will not fail to learn something.
Published on 26 Nov 1998
5.0 out of 5 stars
a landmark book in history of teaching
If this book were taught in every high school class, as it should be, most politicians would have a very hard time in fooling people the way they do.
Published on 2 Nov 1998
5.0 out of 5 stars
a landmark book in history of teaching
If this book were taught in every high school class, as it should be, most politicians would have a very hard time in fooling people the way they do.
Published on 2 Nov 1998
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