I give this book 5 stars not because it is definitive and "correct," but because it proposes an exciting new tack in the philosophy of mathematics. This deeply fascinating book would have been a major addition to that philosophy, a most noble subject, were it not that the authors know little about it. For starters, they do not appreciate the extent to which intuitionists and constructivists have anticipated their attack on what they rightly deprecate as the Romance of (Platonic) Mathematics. Intuitionists entirely agree that mathematics is a human construction serving human purposes; mathematics has no existence apart from this fact.
The philosophy of math has attracted some fine and exciting minds since Frege published his Begriffschrifft in 1879. Around 1900: Russell, of course, but also Husserl. Around 1940: Godel, Quine, Fraenkel, Bernays, Church, Curry, Brouwer, Weyl. More recently: Chihara, Boolos, Parsons, Resnick, Maddy, Shapiro, Detlefsen, Hartrey Field, Burgess, Rosen, Putnam. Regrettably, Lakoff and Nunez appear to have assimilated almost none of this literature.
This cognitive business will eventually have to interact with logic and Ed Zalta's formal theory of abstract objects.
Mathematicians reviewing this book in professional journals, as well as a number of Amazon reviewers, have complained of serious failings in L&N's understanding of some mathematical points. Lakoff argues that the errors found in earlier printings of WMCF are now corrected. On verra...
Readers should keep in mind that Lakoff is a linguist who made his reputation by linking linguistics to cognitive science and the analysis of metaphor. Nunez is a product of the Swiss school of mathematics as grist for cognitive psychology, founded by Piaget. Neither is well-trained in logic, the philosophy of set theory, the axiomatic method, metamathematics, nonstandard analysis and the ontological presuppositions of calculus, the derivations of number systems, and so on.
This book builds on two earlier fine books by Lakoff, his (1987) "Women, Fire and Dangerous Things" and his (1999 with Mark Johnson) "Philosophy in the Flesh.". Both books are very far from academic writing at its worst, but their probing analyses of metaphor, Image Schemata, and other concepts from second-generation cognitive science are not easy. Lakoff (1987) was fascinated by some technical ideas of Putnam's, about which WMCF is unaccountably silent. Lakoff and Johnson contains philosophical riches (thanks to Johnson, a significant contemporary philosopher) that I miss in WMCF. Lakoff and Nunez rightly invoke the authority of Saunders MacLane in support of their position. The authors acknowledge Reuben Hersh very warmly, but do not seem acquainted with his (with Philip Davis) "The Mathematical Experience." WMCF does not cite J R Lucas's "The Conceptual Roots of Mathematics" at all.
Nunez has devoted much of his career to thinking about the foundations of analysis, the real and complex numbers, and about what he calls the Basic Metaphor of Infinity. These topics, worthy though they be, form part of the superstructure of mathematics. The efforts of cognitive science should, I submit, be redirected to the foundations thereof. Now Lakoff and Nunez do pay a fair bit of attention early on to logic, Boolean algebra, and the Zermelo-Fraenkel axioms. And they do linger a bit over group theory. But logic, set theory, number systems, algebra, relations, mereology, topology, and geometry, more or less in that order, should have been the primary focus of their investigation.
I sense that many working mathematicians resist the approach and conclusions of Lakoff and Nunez. This situation is to be regretted. Mathematics has become an extremely powerful toolbox for the mind. Logic and abstract algebra have much to offer to the social sciences and humanities. But communicating these riches to the wider community has proved difficult, and the problem is worsening. For instance, it seems that set theory has vanished from the school curriculum. My students tell me they do not even hear the word "set" spoken until their second year at university. (I learned the core of intuitive set theory around age 12 in the 1960s, and the power of set theoretic metaphors has delighted me ever since.) Even something as basic as first order logic is nowadays learned only by the more technical philosophy majors, and by a small subset of math majors. Hence only a few specialists learn more than calculus, applied statistics, differential equations, and a bit of linear algebra. Just how many persons with a university education know what an equivalence class is? A partial order? A morphism? What it means for a set of axioms to have a model? It is my hope that the cognitive approach to mathematics will suggest improvements to the toolbox of abstractions, and better ways to communicate that toolbox to nonspecialists.