Book Review: Halmos’s “I Want to Be a Mathematician”

Paul R. Halmos, who worked on fields from ergodic theory to algebraic logic and who authored the textbooks “Naive Set Theory” and “Measure Theory”, details much of his academic life in his autobiography, “I Want to be a Mathematician: An Automathography”
. In this post I shall give a few of my impressions of this book.

Halmos writes almost exclusively on his professional life as a mathematician and provides commentary and opinions on research, supervising, teaching, administrative work, and it even includes a math problem or two. In contrast, he rarely writes about his personal life or his other interests besides mathematics. As a mathematics student however, I never thought this as a deficiency and Halmos’s lively and engaging writing kept me steadily reading until the end.

Much of the magic of this book is the atmosphere of the times. Halmos met many famous mathematicians such as Mac Lane, Chevalley, von Neumann, Littlewood, Kolmogorov, and many others that I’ve not listed; his interactions with them, mostly but not always pleasant, are captivating. Halmos’s autobiography is an honest one with both achievements and regrets included.

Halmos describes his research style and his choice of problems. He makes it clear that the quick, frenetic pace of direct competition was not pleasant, and on a few occasions he felt the need to change his program to lessen the effects of direct competition. Halmos was however, fond of changing fields for he felt that it kept the research impulses strong. His work for instance on making logic algebraic is fascinating. It is also fun to read his view that logical methods in mathematics could not provide radically new solutions, and that any solution in logic could probably be written without; this probably partially spurned his foray into logic through algebras. I feel that he would have certainly changed his mind these days if he had a chance to see the applications of modern model theory to algebraic geometry, though one could hardly have imagined such developments back then.

Along with his many exciting adventures, Halmos also injects plenty of advice for graduate students and young mathematicians. I particularly enjoyed the parts about his own thesis problem under Doob. He emphasises reducing general problems to specific cases and using very specific examples. Even though this is something that most graduate students already know, sometimes its easy to forget the importance of examples, especially in abstract fields when so much theory must be absorbed.

Halmos lived a varied professional life and was dedicated not just to research, but also to administration, teaching, and many aspects of publishing. Although he didn’t always enjoy every part of these more mundane tasks, he seemed to always find something interesting in them and furthermore he believed that as a professional, one should be wholehearted even about the mundane but necessary tasks of a professional. His experimentation with the Moore method and his brief stint as department chair at Hawaii testify to his willingness to participate in all aspects of mathematical life.

I found “I Want to Be a Mathematician” funny, never dull, and inspirational. I highly recommend this book to anyone who would like to learn about some aspects of the life of a professional mathematician and the classical years of 20th century mathematics.

Leave a comment

Fields marked with * are required. LaTeX snippets may be entered by surrounding them with single dollar signs. Use double dollar signs for display equations.