American Scientist

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. George Lakoff and Rafael E. Núñez. xviii + 493 pp. Basic Books, 2000. $30.

This book poses a dilemma for the reviewer. It is an ambitious undertaking by a linguist and a psychologist. Their aim is to launch the discipline of the cognitive science of mathematics. As they state in the introduction, the book does not contain mathematical results, but rather "results about the human conceptual system that makes mathematical ideas possible." They succeed in this endeavor to a considerable extent, but the book (and by extension their argument) is marred by serious mathematical errors.

The first two chapters, "The Brain's Innate Arithmetic" and "A Brief Introduction to the Cognitive Science of the Embodied Mind," present the necessary background in cognitive science. The authors cite research demonstrating that, remarkably, infants have innate mathematical capacities—in particular, the ability to "subitize"—that is, to be able to determine instantly how many objects are in a small collection. For more sophisticated ideas, it is necessary to study the cognitive mechanisms that characterize mathematical concepts. Lakoff and Núñez argue that these are the same ones that characterize ordinary ideas.

This leads to the authors' main thesis—that mathematics develops by means of metaphors. The most basic are the grounding metaphors, which "allow you to project from everyday experiences (like putting things into piles) onto abstract concepts (like addition)." Grounding metaphors include "forming collections, putting objects together, using measuring sticks, and moving through space." Their significance is that they "allow human beings . . . to extend arithmetic beyond the small amount we are born with." For more abstract notions, linking metaphors are required. These link arithmetic to other branches of mathematics, such as geometry. An example is what the authors call Boole's metaphor, which links arithmetic operations to "class" (set theoretic) operations. Perhaps the most important linking metaphor is what they call the Basic Metaphor of Infinity (BMI), which allows one to deal with infinite sets, points at infinity, limits of infinite series, infinite intersections and least upper bounds. These are hypothesized as "special cases of a single general conceptual metaphor in which processes that go on indefinitely are conceptualized as having an end and an ultimate result." The next several chapters develop various manifestations of the BMI: limits and continuity, transfinite arithmetic, and infinitesimals, as well as metaphors attributed to Richard Dedekind and Karl Weierstrass.

This material is followed by two brilliant chapters presenting the theory and philosophy of what the authors call "embodied mathematics." Their theory is consistent with, but goes beyond, the position taken by Reuben Hersh in his 1997 book What Is Mathematics, Really? Briefly, Lakoff and Núñez maintain that mathematics is a product of human beings and is shaped by our brains and conceptual systems, as well as the concerns of human societies and culture. We have evolved so that our cognition fits the world as we know it.

This theory is contrasted with both the theory of transcendent mathematics (also referred to here as the "Romance of Mathematics"—an extreme form of Platonism claiming that mathematics is independent of any beings with minds) and the postmodern position that mathematics is arbitrarily shaped by history and culture alone. These two chapters, which conclude with an eloquent "Portrait of Mathematics," constitute a major contribution to the philosophy of mathematics.

Unfortunately, the book's presentation of mathematics is not up to its presentation of philosophy and cognitive science (notwithstanding the endorsements by several eminent mathematicians on the book jacket). There are many obscure and dubious mathematical assertions, as well as downright errors. The authors assert that "so-called space-filling curves do not fill space." No matter how many times I read their explanation, I couldn't figure out what they mean by this. Their presentation of decimals as "infinite polynomials" leaves me cold. Their explanation of why a Taylor series represents a function is incorrect, and they don't discuss the justification for term-by-term differentiation.

There are a number of inaccurate statements, which could have been caught by careful editing or proofreading. (Many of these errors were evidently introduced by the publisher after the manuscript was submitted.) There are some corrections—including a correction to the incorrect definition of limit on pages 198 and 199—on the book's Web site, which can be found at http://www.unifr.ch/perso/nunezr/.

Other errors are more closely tied to the authors' thesis, and so require more discussion. One of the most striking is what they call the "length paradox." This concerns a sequence of "bumpy curves," all of length π/2, which converge to a segment of unit length. This is not a "paradox," but rather a matter of the length function not being continuous in an appropriate topology, and the authors missed an opportunity in not pointing this out. They do finally say that this is not a paradox, but I can't follow their reasoning. (The fact that the curves are not differentiable is not the issue—the same phenomenon can occur when the approximating curves are "smoothed.") Among other things, they mistakenly assert that "limits are defined precisely for sequences of numbers, not sequences of curves."

Another serious mistake occurs in the last section of the book, which concerns the equation e^πi + 1 = 0. As the authors point out, it is necessary to explain what a real number raised to a complex power means. After a quick tour of analytic geometry and trigonometry, a discussion of the number e, the imaginary unit i (with multiplication by i conceptualized as rotation) and the elements of differential calculus, they arrive at the crucial equation e^yi= cos y + i sin y. This is "proved" by showing that both sides have all derivatives the same at 0 (and hence the same Taylor series). But this argument is circular, since e^yi hasn't been defined. The correct route is to define e^yi by means of the Taylor series for the exponential function (so the equation follows immediately by comparing coefficients in the Taylor series.) This is an instance of the principle (enunciated by John Kelley) that a good theorem becomes a definition.

The latter discussion takes almost 70 pages and is presented as "A Case Study of the Cognitive Structure of Classical Mathematics." The aim was not only to prove that the equation holds, but to explain what it "means." It seems to me that the authors were not successful in demonstrating the role of metaphor in the understanding of the equation.

It may be that metaphors don't play a central role in formulating more advanced mathematical concepts, or that if they do, they will need to be of a different nature than those used in more elementary mathematics. Mathematical concepts, once they are developed, acquire a life of their own and are dealt with directly. It's difficult for me to conceive of a metaphor for a real number raised to a complex power, but if there is one, I'd sure like to see it.