xvi Preface

we are a dying breed, that the very reproduction of mathematics as

a social institution and a professional community is under threat. I

approach the problems of mathematical education from this view-

point which should not be easily set aside: what kind of mathemat-

ics teaching allows for the production of future professional math-

ematicians? What is it that makes a mathematician? What are the

specific traits which need to be encouraged in a student if we want

him or her to be capable of a rewarding career in mathematics? I

hope that my observations and questions might be interesting to

all practitioners and theorists of general mathematical education.

But I refrain from any critique of, or recommendations for, school

mathematics teaching.

Alexandre Borovik,

aged 11

The unity of mathematics means that there are no

boundaries between “recreational”, “elementary”, “under-

graduate”, and “research” mathematics; in my book, I freely

move throughout the whole range. Nevertheless, I try to

keep the book as non-technical as possible. I hope that the

book will find readers among school teachers as well as stu-

dents.

In a few instances, the mathematics used appears to be

more technical. This usually happens when I have to re-

sort to metamathematics, a mathematical description of the

structure and role of mathematical theories. But even in

such cases, mathematical concepts are no more than a pre-

sentation tool for a very informal description of my observa-

tions.

Occasionally I could not resist the temptation to include

some comments on matters of my own professional interest;

however, such comments are indicated in the text by smaller print.

Photographs in this book

I come from childhood as from a homeland.

Antoine de Saint-Exupery, ´ Pilot de guerre

I tried to place in the margins of the book a photograph of

every living mathematician/computer scientist/historian of math-

ematics/philosopher of mathematics/scholar of mathematics men-

tioned or quoted in the book. The catch is, I am using childhood

photographs. In my book, I write a lot about children and early

mathematical education, and I wish my book to bear a powerful re-

minder that we all were children once. I hope that the reader agrees

that the photographs make a fascinating gallery—and my warmest

thanks go to everyone who contributed his or her photograph.