Customer Reviews

Prelude to Mathematics

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W. W. Sawyer argues that the pleasure given by a unifying discovery is greatest when a person has struggled with masses of undigested information in an old form. In support of this thesis he reveals unexpected relationships and interdepencies between various topics that might at first seem entirely independent and unrelated.

Despite the wide range of mathematical topics, Prelude to Mathematics only assumes that the reader remembers some basic algebra, geometry, and trigonometry. Calculus is not required. The reader is generally free to skip around; Sawyer has indicated where two chapters are more closely linked and should be read in sequence.

This short book consists of two largely independent sections. The first five chapters (about 60 pages) - On Beauty and Power, What are the Qualities of a Mathematician, Pattern in Elementary Mathematics, Generalization in Elementary Mathematics, and On Unification - provide a general overview of mathematics and mathematical thought.

Chapters 6 through 14 examine more advanced topics often not encountered in lower level mathematics courses.

I especially enjoyed Sawyer's overview of Projective Geometry and its companion chapter, Apparent Impossibilities. Sawyer's discussion of matrices from the perspective of coordinate transformations (rotations, reflections, and stretches) was surprisingly effective. Determinants are traditionally taught before matrices; Sawyer deliberately reverses the order.

Also, the chapters titled On Transformations and On Groups were quite good. The three chapters titled Non-Euclidian Geometries, Algebra without Arithmetic, and Finite Arithmetics and Geometries are good, but perhaps a little dated.

Prelude to Mathematics was first published in 1955 and reprinted as an inexpensive soft cover Dover edition in 1982.

Looking beyond Prelude to Mathematics, I highly recommend Foundations and Fundamental Concepts in Mathematics by Howard Eves for the chapters on non-Euclidian geometries and abstract algebras. Richard Courant's classic text, What is Mathematics? - An Elementary Approach to Ideas and Methods, is another good choice, however, it is a little more advanced.

For the reader intrigued with Sawyer's discussions of matrices, transformations, and groups, I suggest two inexpensive Dover editions: Matrices and Transformations by Anthony Pettofrezzo and An Introduction to Matrices, Sets, and Groups for Science Students by G. Stephenson.

the material in it was over my head, but I had forgotten about it. What a treasure!! MATHEMATICIAN'S DELIGHT is more famous, I believe, but this book is filled with wisdom. Sawyer introduces finite geometries and group theory in simple prose. He figures out where Hall and Knight came up with the ideas for some of their exercises in their (famous) HIGHER ALGEBRA text. He introduces the hypergeometric function as a generalization of power series expressions of functions (something I feel I should certainly have heard of before reading about it here.) I can't imagine anyone with an interest in mathematics not finding something in this book to make him or her say "Hmmm..." or even make him or her pick up a pen and do some figuring.
Highly recommended.

This is the book that really got me interested in mathematics. I had never thought that a math book could be so engrossing. I finished reading it in a couple days and i immediately seeked out the author's other books. And the quality of the other book are of the same level as this one. It is a shame that the author's other books are mostly out of print. What i appreciate most about the book is that the math concepts are always are related to where it came from. The part on series is a small gem, and the book is full of ones like that. Without having met the author, he is in my mind certainly one of the best math teachers ever. (George Polya is another). Thank you, Mr Sawyer.

Prelude To Mathematics covers many aspects of the field of mathematics including modular arithmetic, non euclidean geometry, projective geometry, matrices and determinants. In addition the author discusses common themes in mathematics including pattern, generalization and unification. Although the types of mathematics are introduced in different chapters the author draws them together several times in the book making for a unified approach. I learned a lot from this book. It is the type of book you want to reread because there is so much information in it.

Well, this really is a terrific book. Lots of simple ideas very well explained. He spells out the notion of pattern quite well, and what it means in terms of mathematics. It seems many modern authors just state that math is about patterns and leave it at that. He seems to get to the point on many subjects that a reader has in the back of his mind as he reads about the material.

I'm not sure this book is really intended for the layman. I'm sure many could handle it. Certainly the material is kept at a very introductory level, but he does seem to be talking to the teacher quite often, and some times to the budding mathematician.

My Dover edition doesn't have an index. Too bad. Hard to believe there wasn't one, or should be one. A little bit of laziness on Dover's part?

I have one bone to pick about a short piece of material. He claims Del Square V is the most widely found pattern in nature,
pp. 13-14. He mentions its applications in physics, magnetism, electrostatics, radiation, waves, etc. I'm sure he's thinking of Del dot Del of V. Oddly, he never describes to the lay reader what it actually is. It seems out of place with the material in the rest of the book.) Anyway, he does provide a foot note, which in part says, "It expresses in symbols the law that the value of V at any point equals the average value of V on a sphere with its center at the point. That law treats all points and directions alike, and is the simplest law that does so." Good luck to the layman on that one.

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I got Sawyer's book used in a book store and for 15 years I have used it's wisdom in my work on non Euclidean geometry, fractals and group theory. I just got " Elliptic Curves" by Mc Kean and Moll. For once a Cambridge book that doesn't eat dust! It is so hard to read papers by Yau that one gives up! Most Modern mathematics is written by people who can't communicate... Sawyer is a guide for writers of mathematics... how to do it well!

Please also read Sawyer's Mathematician's Delight. My last review was more appropriate for that book, as it refers to the infinite series example. But this also a superb book. I like M.D. slightly better because of its narrative bent.

Some of the topics presented were interesting, however I wish more topics had been covered rather than being so wordy and devoting whole chapters to one topic such as matrices. Overall, I was not impressed and had to force myself to read to the end. I found the presentation in general dry and uninspiring most of the time. Some math books accessible to the non-mathematician make you go "Wow!" at some of the material they present. This book was not one of them for me. Perhaps this is because it is older. The book does seems to succeed in giving non-mathematicians a flavor for some of what appeals to mathematicians and that's all it does. There seems to be more focus on the abstract rather than concrete and practical aspects so if one likes math more for the "beauty" of it's patterns, etc., then this book will be more appealing. In my opinion, it even fails in that aspect, however, since there is NO material on number theory which seems like an oversight as it plays such a large part in pure mathematics. Some number theory, with a few of Ramanujan's contributions thrown in, may have made this a better book. I much preferred Mathematical Mysteries by Calvin Clawson or The Enjoyment of Mathematics by Rademacher and Toeplitz, for example.

You can SEE and FEEL the MATH and the SOUL of the author