* * * * * Insightful and Provocative, March 22, 2002

Reviewer: Michael Wischmeyer

"The study of mathematics is apt to commence in disappointment."

"One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the textbooks with the solutions of equations."

In discussing Descartes' coordinate geometry, Whitehead states, "Philosophers, when they have possessed a thorough knowledge of mathematics, have been among those who have enriched the science with some of its best ideas. On the other hand, it must be said that, with hardly an exception, all remarks on mathematics made by those philosophers who have possessed but a slight or hasty and late-acquired knowledge of it, are entirely worthless, being either trivial or wrong."

"Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle - they are strictly limited in number, they require fresh horses, and must only be made at decisive moments."

"The really profound changes in human life have all had their ultimate origin in knowledge pursued for its own sake."

Alfred North Whitehead, a remarkable British mathematician and philosopher, enlivens his look at the fundamental ideas underlying mathematics with provocative observations. Nonetheless, Whitehead does not avoid mathematics while trying to explain mathematics. While this book is clearly for the layman, it may occasionally require some effort. An Introduction to Mathematics is delightful, insightful, and intellectually stimulating.

Whitehead argues that mathematics is an abstract science that is primarily concerned with generality, not specificity. In trying to master the techniques and mechanics of arithmetic, algebra, and geometry, many students fail to recognize the fundamental ideas. They become lost in a murky fog of details.

I found myself surprised by Whitehead's insightful explanations of familiar topics like variables, constants, and simple algebraic equations. I know math. But I now recognize that I had not really given sufficient thought to some very basic concepts. Just a few pages into this little book I was actually looking at some familiar concepts from a very different perspective.

Later discussions on mathematical symbolism, imaginary numbers, conic sections, trigonometry, and infinite series move more slowly and may require rereading. But the insights gained will more than offset any additional effort.

Whitehead occasionally digresses to discuss the act of mathematical creation. He agrees with the poet Shelley who compared the discovery of "some great truth" to the slow snowflake by snowflake accumulation that leads to an avalanche. While not discounting the role of genius, Whitehead sees breakthroughs in mathematical thought, often as unexpected as an avalanche, the natural result of the accumulation of knowledge through the centuries.

Whitehead's small book could serve as the basis for a short class or tutorial for high school students (or perhaps even for humanities majors with less than fond feelings for mathematics.) An Introduction to Mathematics is an effective counterbalance to standard textbooks that focus too much on technique, manipulation, and mechanics. Five stars.