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- What Is Mathematics?: An Elementary Approach to Ideas and Methods
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ByMichael Wischmeyeron 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 in 1941. New editions appeared in 1943, 1945, and 1947. My soft cover fourth edition by Oxford University Press is in its 12 printing.

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.

"What is Mathematics?" is not a new book. It was first published in 1941. New editions appeared in 1943, 1945, and 1947. My soft cover fourth edition by Oxford University Press is in its 12 printing.

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.

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ByPalle E T Jorgensenon 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.

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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ByAntonio Villarroelon August 3, 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.

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ByAry Armando Perez Jr.on 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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ByA customeron July 24, 1998

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.

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ByAmazon Customeron December 9, 2002

This is an interesting and wide ranging book. In the main it presents, develops and explains it's ideas very well, although I did not always find it, as one reviewer, a mister Albert Einstein described it, "easily understandable". I have two minor complaints about this book:

1) Print quality

For no apparent reason the text size varies occasionally, and in places the printing is slightly blurred, so that sometimes the subscripts and superscripts on formulae are illegible. Perhaps they skimped on typesetting costs by photoreproducing formulae from the original printing?

2) Incompleteness

If you bought this book because the front cover says "...representation of the fundamental concepts and methods of the whole field of mathematics" (another A.E. quote) you may be disappointed to find this is not the case. Trigonometry, for example, is not discussed, except where it crops up in other topics such as applying calculus to trig functions.

1) Print quality

For no apparent reason the text size varies occasionally, and in places the printing is slightly blurred, so that sometimes the subscripts and superscripts on formulae are illegible. Perhaps they skimped on typesetting costs by photoreproducing formulae from the original printing?

2) Incompleteness

If you bought this book because the front cover says "...representation of the fundamental concepts and methods of the whole field of mathematics" (another A.E. quote) you may be disappointed to find this is not the case. Trigonometry, for example, is not discussed, except where it crops up in other topics such as applying calculus to trig functions.

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ByKersi Von Zerububbelon September 11, 2001

This book will give you a superb introduction to basic mathematics culminating in the CALCULUS. The topics and manner of presentations are excellent. I have the 1978 edition that I still use to much benefit. Things such as numbers, matrices, algebra and trig are introduced in rapid but detailed segments. If you have been away from mathematics for a while you will soon get drawn into the text and the exercises. If you are into math today this will serve as an excellent review and perhaps give you a gem or two. However, if you have been put off by math in the past you may want to approach with caution. For even though the pace is within speed limits the text does expect a good effort to reap the rewards. I recommend this book for anyone interested in the theory behind mathematics. A real jewel for your library and personal enjoyment. Just superb!

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ByAlok Govilon January 31, 2002

A "closer" look at mathematics: Book ends where all other books on Mathematics start. With this book, you look at all of basic mathematics much more deeply. It does not help you how to solve specific maths problems, it tells you why were those problems were solved the way they were by the mathematicians.

A unique blend of actual mathematics, philosophy, and history of mathematics. It goes well beyond defining 'what' is mathematics; Actual mathematics is there in the book.

Example: You know (-1)(-1)=(+1) but never knew that there's 'more' to it -- That it sounds so right yet it doesn't have to be right! With this book, you look at all of basic mathematics much more deeply.

A unique blend of actual mathematics, philosophy, and history of mathematics. It goes well beyond defining 'what' is mathematics; Actual mathematics is there in the book.

Example: You know (-1)(-1)=(+1) but never knew that there's 'more' to it -- That it sounds so right yet it doesn't have to be right! With this book, you look at all of basic mathematics much more deeply.

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ByA customeron October 23, 1999

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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ByGustavo de Oliveiraon September 6, 2002

A very interesting exposition of some of the main branches and ideas of mathematics. This is a book for beginners and experts, students and professors. The authors exposes number theory, algebra, geometry, topology and calculus. (For the last topic I recomend the great book of Courant and Fritz, Introduction to Calculus and Analysis.) The mathematical concepts are introduced and motivated by real problems; it seems to me very close to applications. I have been learning much things with this book. It is very interesting and I recomend it for all people that want to read about mathematics.

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