Where Mathematics Comes From, written by George Lakoff and Rafael Nunez, published by Basic, and reviewed here by John Allen Paulos.

Books on the philosophy of mathematics or mathematical pedagogy are frequently rather tiresome, rehashing standard positions and developing, if the reader is lucky, a nuance or two. Where Mathematics Comes From is an exception, refreshing both in its obliquely revealing approach to these issues and in its fascinating, sometimes iconoclastic conclusions. Contending that mathematics arises out of ordinary cognitive processes, linguist George Lakoff and psychologist Rafael Nuñez deliver a body blow to Platonic notions of mathematics and indirectly suggest broad new avenues of educational research.

From a rather puny innate endowment - an ability to distinguish objects, to subitize (recognize very small whole numbers at a glance), and to add and subtract whole numbers up to 3 - human beings extend their mathematical powers via a complex collection of neurally based spatial relations concepts, image schemas, and, especially, cognitive metaphors. Our primordial experiences with vertical orientation, physical force, and bodily movement lead us to form more complicated idea complexes and to internalize the inferences associated with them. The notion of a conceptual metaphor is familiar from Lakoff's earlier work, particularly The Metaphors We Live By, which underscored the ubiquity of metaphors in our everyday understanding. To grasp abstract concepts, we project embodied or sensory motor reasoning onto them. Affection, for example, is frequently understood in terms of physical warmth: “"She was cool to him”; "he gradually warmed to her"; "they had the hots for each other."

In mathematical terms, Lakoff and Nunez define a metaphor as a mapping from a source domain, which is familiar, to a target domain, which is less so. This correspondence also preserves inferences. That is, statements linked in the source domain are mapped onto similarly linked statements in the target domain. They begin their discussion of mathematical metaphor by discussing at length the four basic grounding metaphors of arithmetic. The first maps ideas involving object collections onto arithmetic. The size of a pile of bricks, for example, suggests a number, and a bigger pile suggests a greater number; our understanding of the notion arithmetical closure. The source domain - the pile of bricks - is an object collection, and the target domain is arithemetic. This the first of four basic grounding metaphors for arithmetic. Another is that of the measuring stick. The length of a physical segment is associated with the size of a number, and so on. Similar metaphorical correspondences exist between arithmetic and motion.

Throughout the book the authors attempt to demystify mathematical thought, stressing that the source of mathematical ideas is not radically different from the source of other, more commonplace notions. They point out, for example, that our understanding of algebraic variables is similar to our understanding of pronouns. "Whoever did this was sick" should be compared to "If X + 2 = 7, then X = 5". Contrariwise they show that misunderstanding can also flow from these prosaic notions. In particular, the root of some of our mathematical mystification is the numbers/things metaphor, which leads to Platonism and our usually unarticulated belief that numbers are entities that are "up there" somewhere. Lakoff and Nuñez are intent on debunking this belief and what they term the "romance of mathematics," namely that mathematics is abstract and disembodied, yet exists independently of us; that mathematical objects and truths are transcendent; that proofs allow us to discover these transcendent truths; and that to learn mathematics is to learn the language of nature.

The book requires a bit more mathematics than many general readers are likely to possess, but one of its pleasures is that it treats numerous areas of mathematics and doesn't skip over all the details. In the more technical second half of the volume, the authors deal with with infinity and the many applications of the basic metaphor of infinity (BMI), proposing that the idea of an actual (not merely potential) infinity derives metaphorically from the notion of the result of a process. Not surprisingly, we conceptualize the result of an infinite unending process in analogy to the result of a completed process that does have an end. The authors hypothesize that all cases of actual infinity in mathematics derive from different and often nonobvious applications of the BMI. Transfinite cardinal numbers and ordinal numbers, for example, stem from quite distinct uses of the BMI, one having to do with infinite collections, the other with infinite lists.

In one of the most interesting and compelling sections of the book the authors describe the cognitive basis of infinitesimal numbers in terms of conceptual metaphors. Many mathematicians consider infinitesimal numbers -- non-zero numbers smaller than 1/N for every whole number N to be "non-rigorous" at best, but the idea is a natural one and we understand it metaphorically in terms of the everyday notion of a speck without discernible width. Leibniz used the metaphor of an infinitesimal, as did mathematicians for almost two hundred years, to develop calculus and its offshoots. In contrast, Newton had to invoke a mysterious new process of taking the limit in his development of the subject. To use this process, we must go through a rather counterintuitive dance. If interested in how fast we're going at exactly three o'clock, we take our average velocity (distance divided by time) over shorter and shorter time intervals around three o'clock and "in the limit" we get the instantaneous velocity. Newton's notion of limit was rightly criticized by Berkeley and others who pointed out that either a time interval has length zero in which case division is impossible, or it does not and we don't have the instantaneous velocity. This logical lacuna wasn't filled until the 19th century by Cauchy, Weierstrass, and others, and even today students are often baffled by the traditional definition of limits. More in accord with our intuitions, infinitesimals work more naturally since one can actually divide by these non-zero specks, and finding limits is reduced to a kind of arithmetic.

Although Lakoff and Nuñez emphasize the contingent, sometimes cultural origin of mathematical ideas and disciplines, they reject the postmodernist view that mathematics is merely a cultural artifact. Perhaps because they want to distance themselves from excesses frequently associated with postmodernism, their writing, despite its focus on metaphor, is devoid of rhetorical fluorish. They hardheadedly avow their appreciation of the precision, universality, and abstraction of the subject., and they grant that there are such things as mathematical discoveries (as opposed to inventions), that are possible once mathematical concepts are established within a community. Yet do they really succeed in banishing all Platonic notions, which they compare to religious ideas? Although few people can continue to believe in a literal Platonic heaven where numbers, functions, curves, probabilities, and other mathematical characters cavort timelessly, a more sophisticated Platonism is still defensible, especially with respect to the integers and theories associated with them.

Tellingly Leopold Kronecker, a 19th century contemporary of Georg Cantor, used religious/mathematical language to make the same distinction between what is eternal and what is not. He famously pronounced that God created the integers and that everything else was man-made. The sheer variety of set theories, conceptions of real, granular, and hyperreal numbers, and alternative geometries, algebras and logics do suggest their non-transcendent nature, and the authors make "atheistic" disbelief in their Platonic residence more plausible. On the other hand, the independent existence of the integers seems to call for a more agnostic stance. Arithmetic may, for example, be transcendent in the sense that any sentient being would eventually develop the metaphors that ground it and be led to its truths, which can thus be said to inhere in the universe. Other patches for a quasi- Platonism are also possible.

Whatever one's views on the nature of mathematical entities and truths, however, Where Mathematics Comes From is salutary in its emphasis on the human origin of mathematical concepts. Careful study of the conceptual metaphors that give meaning to mathematical ideas should be quite useful pedagogically, and it is in this area, rather than the philosophy of mathematics, that this provocative book's impact will be greatest. A better realization of the sometimes unconscious, usually mundane sources of mathematical ideas can only help us learn and teach. All in all, the book adds body heat to the cold and beautiful abstraction of mathematics.