Rating: 
Summary: For mature thinking in mathematics
Review: 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.
Rating:
Summary: A book on mathematical ideas to look beyond the formalism
Review: 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 expose number theory, algebra, geometry, topology and calculus. (For the last topic I recommend 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 many things from this book. It is very interesting and I recommend it for all people who want to read about mathematics.
Rating:
Summary: Extraordinary Book. Belongs on Your Bookshelf.
Review: 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.
Rating:
Summary: Timeless.
Review: 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.
Rating: 
Summary: Pretty good
Review: 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.
Rating:
Summary: Pretty good
Review: 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.
Rating:
Summary: classic
Review: 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.
Rating:
Summary: Lucid!!!
Review: The title suggests a topic more on the philosophy
of math rather than math proper, but it's all real math, and superbly
done!
Rating:
Summary: A classic that will challenge and inspire.. A MUST HAVE!!
Review: 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!
Rating:
Summary: A classic that will challenge and inspire.. A MUST HAVE!!
Review: 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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