This book, originally published in 1942 by the Council on Books in Wartime as a 3-3/4x5-3/8” softcover edition for G.I.s (according to the introduction by Barry Mazur), was copyrighted in 1944. Lieber sets out to show a number of things about math: it works better than a guess or a tedious working out of a solution; it has a philosophy (Interntionalism and Democracy―such concepts, along with Abstraction, Generalization, and others are always capitalized and sometimes the whole word is in caps); its generalization and abstraction tools are useful in all thinking. Some traditional math has remained very useful, but some has been superseded for certain purposes, as calculus replaced traditional math in dealing with dynamic quantities. Here as often in book there is a Moral: choose carefully among the traditional and the new. Thus much (along with a summary) is Part I of the book: “The Old.”
At the end of Part I Lieber has a chapter trying to show how one idea of calculus works. It didn't work for me on the first try. Her tone throughout is as if lecturing a child, though her intention may be merely to be clear. And peculiarly, the book's print lines are not justified, but writen in phrases on separate lines, which, she says, “facilitates rapid reading.”
In Part II, “The New,” Lieber tries to show how one might work within a non-Euclidean geometry―this one with only 25 points. Then she gives an example of a “finite algebra.” The point seems to be that T. C. Mits needs to understand 2 + 2 does not always equal 4 and parallel lines might look different in an alternate geometry. But they are useful innovations. Finally, she says that “scientific predictions are a triumph of CLEAR THINKING even if they are not THE ABSOLUTE TRUTH.” Somehow this is supposed to make us more tolerant of Modern Art―more tolerant in general. ( )