Is math real? How simple questions lead us to mathematics' deepest truths

"Math might seem like it's about getting the right answers, but really it's about the process of discovering," according to this invigorating philosophical take on the field. Mathematician Cheng (The Joy of Abstraction) explores how such questions as "how many sides does a circle have?" and "why does --(--1)=1?" reveal surprising profundities about math. She suggests that situations in which one plus one does not equal two (one pile of sand placed on another makes for one pile) shows how numbers are ways of abstracting and simplifying the world that require individuals to decide what to count (piles of sand) and what to ignore (the individual grains in each pile). Classrooms, she laments, typically shun such modes of inquiry in favor of instilling "rigidly imposed rules," contrary to the "point of math," which, she argues, is "learning how to decide what counts as a good answer when there is no answer key." Cheng has a talent for making mathematical discussions accessible, and her wide-ranging analysis leads to some surprisingly weighty conclusions, as when she argues that expecting students to accept mathematical rules without question sends the message that truth comes from authority, making it nigh impossible to reason with students "because their beliefs aren't based on reason; they're based on authority." It adds up to a stellar meditation on the nature of knowledge and math. (Aug.)

An abstract if oddly entertaining foray into the more philosophical realms of mathematics. A noted popularizer of mathematics, Cheng, the author of Beyond Infinity and How To Bake Pi, works at the frontiers of the discipline in an arcane area "called category theory," which "doesn't involve numbers and equations at all." If the thought of math without numbers makes your head hurt, the author's latest book will be a constant challenge. Math is real, she tells us, in much the same way that Santa Claus is real: as an idea. Thus, as she puts it, it's entirely possible that another idea can come into play, namely that 1 + 1 does not equal 2; the question then becomes not "What is 1" or "What is 2," but instead, "What is a world in which 1 + 1 = 2?" Given that math, in concert with physics, admits the possibility of an infinite number of worlds, or dimensions, a world where 1 + 1 = 1 isn't out of the question. Our world gives the answer of 2 because that's the abstraction we agree on, just as we agree (for the most part) on the laws of logic--and that's a key idea, for, as Cheng says brightly, "Mathematics is the logical study of how logical things work." The strict rules of logic can, of course, make a person's head hurt, too; one has only to think of Zeno's paradox, wherein neither the tortoise nor the hare actually wins a race because "the sum doesn't converge." Some of the author's examples take the form of equations, and while it helps to be numerate, the numerophobic shouldn't shy away from digging in. Despite her provocative title, others are fun examples from the very real world, such as using a recipe for mayonnaise to discuss the process of commutativity. For the budding mathematician in the house, to say nothing of lovers of puzzles and enigmas. Copyright (c) Kirkus Reviews, used with permission.