Customer Reviews

What Is Mathematics? An Elementary Approach to Ideas and Methods

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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.

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.

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!

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.

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!

I always liked math as a child and was quite talented, but I absolutely hated the way math was taught in school. For a long time I could always grasp the necessary concepts intuitively, and so I didn't have to bother with the books we were supposed to be working with (my teachers were frustrated over my not doing any "exercises", but I got away with it since I aced the exams). But when it came to high school and a bit more demanding math (mainly calculus), I couldn't do it all by pure intuition and since I couldn't deal with the math textbooks, I fell behind. Eventually I developed an outright aversion for math, confusing my life-long love for the subject with my intense dislike of how it was taught in school.

I kept looking for a better way to learn math - I just knew there had to be one. University textbooks was often better structured, but they presumed a lot of skills that I didn't have yet. Far too many years later I randomly came upon this book, and in it I found everything I had been looking for all along, namely clear and concise discussions of mathematical concepts.

In school, the focus was always on exercises, with no clear explanation of the concepts involved. In fact, discussion of mathematical concepts was clearly avoided, even when it would have seemed quite natural. The general idea seemed to be to get kids to use mathematical concepts more or less blindly, and thereby "learning" them without having to think about them. For me this was just utterly perverse and unspeakably frustrating. Being intensely interested in understanding things, but strongly averse to mindless repetition, more than anything I felt like I was being punished, expected to learn how to use math, but deviously kept from actually understanding what I was doing. In practice, I could absolutely not do the exercises without first grasping the concepts, and once I knew the concepts, the type of exercises provided were meaningless.

When I found the book by Courant, I knew after only a few pages that I was home. Here were rigorous explanations and discussions of all the basic concepts of math, beginning on the first page with a discussion of the concept of number, moving on through the book to include the foundations of number theory, geometry, topology and calculus. There are also some exercises in this book, but they are not in any way the main focus, and they are quite well devised. Despite struggling with an almost pathological aversion to exercises in mathematical textbooks, I quite enjoyed them!

If you are an intelligent person who has given up on math, despite having a vague sense that you really should be enjoying it, perhaps you should take a look at this book. Doing so is very easy since large parts of it are viewable for free through Google Books.

Covers a lot of different branches of mathematics in depth but does not require a significant mathematical backround. This is good for beginners; you can even learn the basics of calculus, and in a much better way than those "calculus made simple" types of books. I particularly liked the sections on algebraic number fields, geometric constructions, Louisville's construction of transendental numbers, a "probabilistic proof" of the Prime Number Theorem, and the whole section on functions and the Calculus. Actually, the whole book was great. Get it.

I read this book while in high school, before enrolling in college. I was captured by the beauty of the subject. Every concept seemed alive: a triumph of imagination, intuition and intelligence. I chose Physics and Mathematics thanks to this book. Without exaggeration, this book had a big influence in my life. Still today it has no equals, and is an enjoyable yet challenging reading for any reader armed with a good hig-school-level math knowledge, and some curiosity.

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!

The book is littered with mangled formulas, mostly due to the fact the minus sign is missing from most formulas. This is completely unacceptable in any math book, but particularly so in a book aimed at beginners, who will probably feel bewildered by the huge amount of nonsensical formulas.

Adding insult to injury, the poor layout of Kindle edition makes the book hard to enjoy. Section headings often appear as orphans at the bottom of pages and many formulas are displayed as small, low-quality images. The low image quality is especially visible when reading the book on a Kindle 2 as opposed to for example the Kindle Mac app. Inlining formula images with running text is particularly annoying, since it deforms the text layout.

I was so annoyed by these problems that I tried to return this book, only to discover that returns are not allowed for digital books.

Do NOT buy the Kindle edition of this book.