The book is exciting, witty, and well worth the time invested in its study. It communicates what it means to be a mathematician.
–The Mathematics Teacher
At last we have a thorough account of the period that runs approximately from the forties to the present day, a period that may go down in history as one of the golden ages of mathematics.
–The American Mathematical Monthly
I Want to be a Mathematician is an account of the author’s life as a mathematician. It tells us what it is like to be a mathematician and to do mathematics. It will be read with interest and enjoyment by those in mathematics and by those who might want to know what mathematicians and mathematical careers are like.
Table of Contents
Reading and writing and ‘rithmetic
A college education
Learning to study
Learning to think
Winning the war
A great university
The early years
The fabulous fifties
How to teach
To Sydney, to Moscow and back
How to do almost everything
Service, one way or another
Index of Photographs
Excerpt: How to do almost everything (p.319)
I was at the height of my powers in the 1960’s—which may interpreted as saying that, having reached and passed the maximum I was capable of, I was over the hill. The height of your powers takes an instant to attain; from then on it’s down the rest of the way. I have all my life with one value system–research is all–and height therefore refers to theorems, or, in a better word, insights.
The three most visible concepts on my mathematical horizon in the 60’s are identified by the words Toeplitz, Cesàro, a quasitriangular.
Toeplitz operators played a big role at the farewell conference in Chicago just before I went to Michigan. I don’t why Toeplitz matrices are attractive, but it is a fact that every mathematician who is introduced to them finds them natural and pleasant. (They are the matrices with constant diagonals: ai, j = ai + k, j + k.) Arlen Brown and I became fascinated by them, rad about them (principally in the papers of Philip Hartman and Aurel Wintner), and started asking questions about them. We got some satisfying insights—we thought we really understood what made the Hartman-Wintner theorems tick, we gave a characterization of Toeplitz operators, and we laid the foundations for their algebraic study. Our paper is old stuff by now (it appeared in 1963), the theory has made much progress since then, but I am proud to have had a hand in it.
The work on Cesàro, operators was another collaborative effort. It was written in a year when my two-man seminar with Allen Shields became a three-man seminar by the addition of Arlen Brown. We were all at Michigan then, and we got together Tuesday afternoons for a couple of hours to drink coffee and talk Cesàro. (Surely everyone knows the Cesàro matrix? It is the one that converts a sequence onto the sequence of averages; its n-th row has the first n entries equal to 1/n and the rest equal to 0.) We looked at boundedness problems and spectral problems for that matrix and for its “continuous” analogues, both finite and infinite. We solved a lot of them, but we were stumped by one: we couldn’t prove that the classical Cesàro operator is subnormal. The proof was found six years later by Tom Kriete and David Trutt; it is a difficult and deep achievement.
About the Author
Paul Halmos is well-known for his research in ergodic theory and measure theory. He is on of the most widely read mathematical expositors in the world.
Paul Halmos's "automathography" is a classic of the genre. First published in 1985, it contains Halmos's memories of his (then) 50-year career as a mathematician, from the 1930s to the 1980s. This is an essentially unchanged reprint of the original Springer edition. The changes are cosmetic: the cover is prettier, it's a paperback, and a few of the photographs have been replaced or retouched.
Halmos's basic approach is made clear in the "overture":
"Sure, I had parents (two) and wives (two, one at a time, the present one for forty years), and cats… I like Haydn, long walks, Nero Wolfe, and dark beer, and for a few years I tried TM. All that is true, but it's none of your business — that's not what this book is about."
Instead, the book is about his life as a mathematician among mathematicians. It shows us a little about how Halmos thinks about mathematics, about what interested and motivated him, and about how he interacted with others. It includes a lot of what might, somewhat uncharitably, be described as "gossip": stories and anecdotes about mathematics, mathematics departments, and mathematicians. Continued...