Lillian R. Lieber must have been a fascinating person. Beginning in the 1930s, she wrote (and her husband Hugh Gray Lieber illustrated) a sequence of "popular" books on mathematics. The books are all written in a unique style: sentences are broken into chunks that are supposedly easier to read and absorb. Like this, from the preface of the book under review:

This is not intended to be

free verse.

Writing each phrase on a separate line

facilitates rapid reading,

and everyone

is in a hurry

nowadays.

The books were full of whimsical illustrations, words in ALL CAPS for emphasis, and moralizing commentaries about mathematics. And the topics ran all the way from quite technical (Galois theory, infinity, and Relativity Theory) to the kind of popular mathematics writing that we find, for example, in *The Education of T. C. Mits,* first published in 1942 and then, in a revised and enlarged edition, in 1944.

"T. C. Mits" is "**T**he **C**elebrated **M**an **I**n **T**he **S**treet" (one can't help but comment that today he'd be "Pits"… though one should note that his wife, Wits, played a role in a later book, *Mits, Wits, and Logic*). His "Education" is really an attempt to argue the power and virtues of mathematical thinking. Along the way, Lieber discusses several interesting bits of mathematics. Some are standard recreational math problems (imagine a belt around the Earth…). Several, however, are quite interesting and unusual: a finite geometry, a finite field, a space that is "naturally" four-dimensional (the parameter space for the collections of all spheres in 3-space, an example that I learned from this book over 35 years ago and have used forever after).

The main goal is to make two arguments: that mathematical thinking is both valuable and (in the long run) useful, and that it models ways of thinking that are applicable more widely.

The first point is, of course, one that is still made in many popular mathematics books and in video programs such as PBS's *Life by the Numbers*. Today, writers in this vein prefer to show snazzy new applications of mathematics, often not saying much about the mathematical ideas themselves and certainly keeping "pure mathematics" well in the background. Lieber is more daring. She gives us the image of a "totem pole" of science, with technology at the bottom and mathematics at the top. (Applied mathematics, science, and engineering occupy the middle levels, and, since there are five levels, each is represented by one of the regular polyhedra!) Lieber's argument then is that each levels uses the products of the level above. As a result, any snazzy new technology depends, ultimately, on the pure mathematics at the top. This "trickle-down" approach to making a case for mathematics is now old-fashioned; Lieber's presentation is as effective as any I've seen.

On the way to making her case for the value of mathematics, Lieber deals with a concern very much in evidence in the 1940s, namely the use of science and technology (and therefore, given the totem pole, of mathematics) in war. Her argument is that while it is true that science and technology can be so used, in their essential philosophies science and mathematics point in another direction.

(It is remarkable that the morality of war itself is not part of the discussion. I think this is because, in the middle of World War II, the awfulness of war itself was more vivid to most people than the issue of whether the war was necessary.)

This leads to the second argument, namely, the claim that mathematical thinking is a useful model for human thinking in general. This is where the book appears most old-fashioned. How often does one see it argued that the way mathematics is done supports both internationalism and democracy? The argument that logical thinking is useful everywhere is easier to make, but Lieber goes way beyond that.

The preface by Barry Mazur captures well my reaction to this side of the book:

The first thing that reading it does to me is to return me to the state of twelve year old, the authors coming at me like a loving, elderly aunt and uncle. They plunk me onto a chair which is a bit too high for my legs to reach the ground, and slide a large glass of chocolate milk and a home-baked cookie in my direction, all in preparation for showering me with exuberant lessons (of life, of math, of the world … stuff like that), each with its concomitant ulterior moral.

Exactly! It's hard to say how today's readers — at least those who did not have such uncles and aunts — will react to that.

Given this "improving" aspect of the book, it's not surprising to learn that it was also published in an "Armed Services Edition" for the use of American soldiers fighting in World War II. (There is an online exhibit dedicated to the Armed Services Editions, from which we obtained the image on the left.) Future science fiction writer Frederik Pohl reports that this edition of *Mits* represented his first exposure to the idea of formal logic.

In his foreword, Mazur comments:

What luck the Council on Books in Wartime had, to be able to offer the troops this educationally uplifting book, speckled with its bright cartoon drawings of T. C. Mits, who — in defiance of (or perhaps oblivious to) the ravening forces of destruction of the age — is often shown sporting a hyperboloid cap and contemplating, cheerily, his place in some vast setting of the mathematical sublime.

Lieber seems to have something of a following. She has a Wikipedia entry, and there is also a web page specifically about her and her husband's work. Perhaps this interest will lead to others of her books coming back into print.

To my mind, this is not the best of Lieber's books (I'd give that to her *Infinity*, at least partly out of nostalgia: it was the first one I read). Still, it's a delightful book that fights the good fight for the discipline we all have learned to love, mathematics. It's great to have it back. I'm planning to ask my calculus students to read it. Maybe we can have an interesting debate on whether mathematics really does serve as a guide to clear thinking in all of life.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME, and the editor of *MAA Reviews*. He read his first Lillian R. Lieber book in junior high, and hasn't recovered yet.