Prelude to Mathematics

by W.W. Sawyer

"No mathematician can be a complete mathematician unless he is also something of a poet." — K. Weierstrass
In this lively and stimulating account, noted mathematician and educator W. W. Sawyer (Professor Emeritus, University of Toronto) defines mathematics as "the classification and study of all possible patterns." It is a broad definition, but one that seems appropriate to the great scope and depth of the topic. Indeed, mathematics seems to have few boundaries, either in applications to practical matters or in its mind-stretching excursions into realms of pure abstraction.
Gearing his approach to the layman whose grasp of things mathematical may be a bit precarious, Professor Sawyer offers a lucid, accessible introduction to the mathematician's cast of mind. Five well-written preliminary chapters explore the beauty, power and mysticism of mathematics; the role of math as an adjunct in utilitarian matters; and the concepts of pattern, generalization and unification as both tools and goals of mathematical thought.
After developing this conceptual groundwork, the author goes on to treat of more advanced non-Euclidean geometry, matrices, projective geometry, determinants, transformations and group theory. The emphasis here is not on mathematics with great practical utility, but on those branches which are exciting in themselves — mathematics which offers the strange, the novel, the apparently impossible — for example, an arithmetic in which no number is larger than four.
Mathematicians will appreciate the author's grasp of a wide range of important mathematical topics, and his ability to illuminate the complex issues involved; laymen, especially those with a minimal math background, will appreciate the accessibility of much of the book, which affords not only a portrait of mathematics as a matchless tool for probing the nature of the universe, but a revealing glimpse of that mysterious entity called "the mathematical mind." Professor Sawyer has further enhanced this new Dover edition with updated material on group theory, appearing here in English for the first time.

Genres: Mathematics, Science, Nonfiction, Philosophy

224 pages, Paperback

First published January 1, 1955

About the author

Walter Warwick Sawyer (or W.W. Sawyer) was a mathematician, mathematics educator and author, who taught on several continents.
https://en.wikipedia.org/wiki/Walter_...

Reviews

[Jerry Balzano · Rating 5 out of 5]

See my review of Mathematician's Delight. Basically, anything written by W. W. Sawyer is pure gold, and worth reading.

[Kyle · Rating 5 out of 5]

Every time I have picked up a W.W. Sawyer book, I have not been disappointed. I mostly regret not having read his books earlier (I started with the last book he wrote, on Functional Analysis in graduate school). Sawyer has a particular mathematical writing style that I find fresh, insightful, and understandable. He uses humor, lots of analogies and pictures, and well-motivated examples to get you to learn a concept. This book is no different, and I plan on going through the rest of the catalog of books written by Sawyer.

This book covers a variety of mathematical topics, from groups to projective geometry, to determinants, and to matrices. You won't be an expert, but you'll have a flavor of what's going on. Sawyer wrote it for people thinking about getting deeper into math, and I think it would work wonderfully there (though I'd like such people to read it and give me feedback), but I have a ton of mathematical experience, and the book shines for me as well. It explains connections between various branches of math, and helps you take a new look at old subjects and just reminds me of the "beauty" (or whatever you want to call it) of math, and learning a math truth for yourself. I particularly enjoyed that Sawyer doesn't dumb anything down. He even talks about the hypergeometric functions (I have an interest in them, since they are basically every function you ever work with in physics) and he explains them without overly complicating things.

There are just a lot of great insights, humorous statements, and wonderful analogies throughout. I am so sad that Sawyer didn't write even more.

[Cheryl · Rating 5 out of 5]

The book is very stimulating, and if read carefully you will find that you evolve with it from chapter to chapter. After completing the book, I was compelled to re-read the first chapter "On Beauty and Power". Here Mr. Sawyer states "This book is about how to grow mathematicians." He uses mathematics as a catalyst to show how the universe is constantly seeking balance..."the solution of one problem leads to another problem to be solved", and how we as citizens of the universe must remain open and be willing to see and to grow.

[Anup · Rating 5 out of 5]

I read this book lying in bed while it was snowing outside. I didn't take notes or write anything. However, I enjoyed this book a lot and learnt a few things from it. It would have been better if I had spent a bit more time reading it. I can always come back and do it again later. I like mathematics and there were many interesting things in this book. It will definitely help me when I read other books on mathematics in the future.

[John Conway · Rating 3 out of 5]

Great wee book.

Parts that I didn’t fully understand but that’s okay.

[Jacob · Rating 3 out of 5]

On general, I really enjoy math books, but I am only giving this one a few stars. The main reasons are its dated-ness (e.g. applications and methods have changed significantly with computers/calculators, encryption, etc.) and that the title and back cover make it sound like a good book for someone with little mathematical background. I would argue that it is geared more for people who already enjoy mathematics and perhaps are thinking about pursuing a degree in the subject. Consequently, the book is too hard for beginners (perhaps expectations of "laymen" with "minimal math background" were higher in 1955?) yet lacking much of the material of interest (e.g. omitted proofs) to more advanced readers.

[Paideia Sofista · Rating 4 out of 5]

Padrísimo libro con un final abrupto que no cierra ni siquiera el último capítulo.

El autor se propone y consigue dar una buena zambullida a quienes —como lo hiciera yo en los 90’s— aspiran a convertirse en matemáticos que los familiarice con le elusiva esencia de las Matemáticas.

Cada capítulo es un pequeño tratado pedagógico sobre algún aspecto propio de dicha esencia. Y Sawyer lo explica magistralmente poniendo manos a la obra en una serie de problemas cuya virtud es mostrar las entrañas de alguna rama de las Matemáticas. Hace una mezcla perfecta de 'paper', libro de texto y ensayo introductorio.

Es un libró matemáticamente bello.

*****

Me tardé tanto tiempo en leerlo porque me sobrevino una racha de mal dormir. Inicialmente avancé bastante bien leyéndolo en mis traslados en trolebús, pero al llegar a la mitad comenzaron los problemas. Con trabajos leía un par de páginas antes de ser derrotado por el sueño.

Sin embargo lo disfruté horrores. Y saldé una deuda con aquel jovenzuelo que soñaba con ser matemático, que, atraído por el título y la descripción de la contraportada, compró en la librería de la UAM-I un libro del que, por mucho que le fascinó, tan solo llegó a leer unas veinte páginas.

[Andrew Schulz]

This was exactly the right book at the right time for me, which is not an experience you can predict for anyone. It covered major branches of mathematics in a few short chapters, nailing down the bases in a few short logical demonstrations, and allowed me to see for myself just how they'd be applicable to modern physics as a whole. As you can tell, I'm a math newbie, but we all had to start somewhere. Sawyer's books are an excellent starting point for anyone.

[Simon · Rating 5 out of 5]

Highly recommended for anyone wanting to understand mathematical thinking processes and how math(s) can become more useful as it becomes more abstract.

[db · Rating 3 out of 5]

Difficult for me to understand

[Giorgi · Rating 3 out of 5]

beginning chapters are more insightful - later ones are more technical
still worth reading

[Douglas Murphy]

that was fun, it's an old book by now but I did find its examples and explanations helpful and, importantly, graspable.

[Debby Slaydon · Rating 3 out of 5]

Hard to follow on a E reader

With an E reader, it was too difficult to flip back and forth among diagrams, new math rules, equations, and results. Being ab engineer, I had a hard time with seeing any value of the new maths beyond “you can” and there might br interesting patterns.

[Ben Gutierrez · Rating 3 out of 5]

This is a bird's eye view of modern mathematics. It touches lightly on a wide variety of things, and is not rigorous. I especially liked the explanation of determinants in linear algebra and the broader sense of how mathematicians think. I felt cheated when I came to the end of the book and realized that my mind hadn't been pushed. Perhaps that was all intentional, since it wasn't Mathematics. It was the Prelude.

[Keith Clark · Rating 2 out of 5]

Rambled too much

[Suka · Rating 4 out of 5]

Felt like I was watching a 3b1b video. Amazing!