This book introduces a new sub-discipline of cognitive science called “mathematical idea analysis”. In brief this sub-discipline seeks to provide a more intuitive understanding of many abstract mathematical ideas by tying those ideas to the often unconscious understanding of our physical environment (embodiment). From the preface:

“Human ideas are, to a large extent, grounded in sensory-motor experience. Abstract human ideas make use of precisely formulatable cognitive mechanisms such as conceptual metaphors that import modes of reasoning from sensory-motor experience. [ … ] The central question we ask is this: How can cognitive science bring systematic scientific rigor to the realm of human mathematical ideas, which lies outside the rigor of mathematics itself? Our job is to help make precise what mathematics itself cannot—the nature of mathematical ideas.” Pg. XII

The book is creative, ambitious, and well organized. For the most part the writing is good, although I thought it became repetitive at times where the authors would start a new section by repeating several ideas from the last section. Some people may like this style - it’s very structured and “scientific” - but it also gets tiring, especially in a 450 page book.

I think overall the book is a mixed bag. The introductory chapters and the summary chapters (on the theory and philosophy of embodied mathematics) were pretty good and I enjoyed them. But the majority of the books middle chapters are focused on a very detailed construction of all the metaphors needed for mathematical idea analysis. There are countless tables making detailed metaphorical mappings from “source domain” to “target domain”. These often seem trivial or obvious and I kept thinking to myself ”Well, someone’s got to do this but I’m not really interested in reading about it!” (I imagine the same thing has been said of Russell and Whitehead’s “Principia Mathematica”, it was an amazing achievement but not great reading). My feeling is that only specialists in cognitive science or mathematical pedagogy would find these chapters useful.

Overall I wouldn’t recommend this book for a general reader wanting to learn more about mathematics or cognitive science since I think there are better books on these subjects. For instance, if you are interested in cognitive science and the idea of embodiment I highly recommend Lakoff’s earlier book “Women, Fire, and Dangerous Things”. If you already own "Where Mathematics Comes From" then reading chapters 1, 2, 15 and 16 should give you a good feel for the book and whether you want to continue or not. ( )

I've never ready anything about cognitive science and as this book is a look at Mathematics from the Cognitive Scientist point of view it was difficult to start. By the end of the book I was pretty enthralled. Any one that has taken some higher level math courses (analysis, abstract alg) should read this book and really think about what they've learned. Anyone planning on teaching higher level math should read this and think about how they teach.

Can be summarized by the dust-jacket slogans "Mathematics is not built into the universe" and "The portrait of mathematics has a human face." Meaty and quite absorbing, even though it goes against my sometime Platonist sympathies.