Where Mathematics Come from How the Embodied Mind Brings Mathematics Into Being

From most of what I've read of this book I've found it very excellent. It provides intriguing ways to think about mathematics and to help us realize how mathematical thinking works. In terms of pedagogical value, this book might surpass all others. Still this book has received much criticism. Some people have commented that this book contains contradictory statements, and consequently that it can't qualify as mathematical. This really doesn't hold in mathematics, as modern mathematics rather consistently come up with contradictory statements, such as the paradoxes of (crisp) set theory. Addtionally, in terms of a logical basis, there doesn't exist any compelling logical reason that anyone must accept the principles of classical logic as true. Consequently, even if this makes proofs more difficult (and that I consider the real issue), mathematically contradictory statements present no real problem. However, this book does characterize (abstract) algebra as relying on the folk notion of essences. The notion of essence basically means that something has an inherent specific nature which determines what it is. The essence of something makes it what it is completely. Abstract algebra doesn't talk about essences. It talks about how something behaves. Consider the field axioms as given by Lakoff and Nunez in their excellent discussion of granular arithmetic. Those axioms do not determine what granular numbers are. The discussion on the few previous pages, MORE OR LESS, tell us what granular numbers are. The axioms tell us how granular numbers behave under certain operations (though by no means under all valid operations or in all conditions). As another point consider the following axioms one could use for a classical logic. Given an operation for intersection '^' and 'v' for conjunction, the following hold: 1^1=1 and 1^0=0^1=0^0 1v1=1v0=0v1 and 0v0. Those axioms do hold for every system of classical logic. But, let's say we talked about a mathematical system (max, min, 1-a, {0, .5, 1}), with '^' standing for 'min' and 'v' standing for 'max', and {0, .5, 1} indicating the possible values of our variables. The above axioms still hold. But, this system does not qualify as a classical logic, since all classical logics only have {0, 1} as the possible values of its variables. Still, our system with (max, min, 1-a, {0, .5, 1}) will BEHAVE like a classical logic when confined to {0, 1}. This is in no way invaldiates this book. I still highly suggest reading it and thinking about its concepts. The book, unlike many other books, realizes that many different systems of logics, and many different mathematics actually do exist and get invented. I would hope that many more mathematicians would someday realize that mathematics consists of a large variety with different rules for its domains, not needing consistent across them.

I could imagine that the authors of this book might reply to an earlier review by explaining how Pi, which is 3.14159..., exists only in the human brain as the notion of a perfect circle. It is the way the human brain describes what does not actually exist. As far as I know, no circle has been found or created where if we were to measure it at the atomic level we would find the circumference divided by the width yielding an infinite series of digits that works out to our pi. Zooming in on the edge of a circle to get infinite precision would not work -- the granularity would give way to decreased edge perfection. Creation is not as clean as the human's ideal circle. And we can't clean up creation and create the perfect circle infinitely accurate down to the quark level. What we can say is that circles -- from human observation -- are best codified in neural pathways by the notion of pi. But pi does not actually exist (and it is only romantic faith that believes that it exists in some world -- a world no one has seen!). In "Where Mathematics Comes From", Lakoff and Nunez defend their thesis that the only kind of mathematics that humans can know are the kind that are known to human minds. Human mathematics is embodied mathematics, and not necessarily representative of some absolute transcendent truth. This non-Platonic way of looking at math should liberate the reader from what the authors call the "Romantic" version of math. Romantic math involves the mathematician casting his symbolic universe into the heavenly realm as if math were a religious expression of eternal norms -- norms that everyone is expected to observe on bended knee (as it were). Such notions of math are not embodied, but transcendent and disembodied (existing outside of humanity). The authors take this romantic view to task (I think rightly so). Cognitive science is one way they arrive at their counter philosophy. And the study of the brain is how they ground mathematics in humanity, as to take it out of the theological realm. In so doing, they work from at least three key ideas: 1) Our life and our experiences (i.e. our bodily existence) shape our knowledge, our structures and our concepts. We don't know transcendent mathematical truth -- we know the math that is knowable by the human brain, which is the embodied mind. This idea that our knowledge conforms to the structure and makeup of our humanity is so basic to the thesis of the book -- and so simple a concept -- that it may be easy to miss its power. This idea frees mathematics from a kind of religious absolutism that has created fear and awe of the subject. Math is not a deity, nor is it the realm of the deity -- it is the domain of the human mind (which is not to deny a real deity, but only to locate math in the only place we know it to exist: in the human). The authors provide convincing reasons why this is true, then they give convincing reasons why we need to get this right. Getting it wrong has

As a person interested in math, physics, philosophy, and cognition, I was delighted to find a book that helps tie these fields together. I've read many popularizations of math history and theory, and this books goes far beyond any of them. First of all, this book is NOT a popularization, nor is it a book on math. It is a serious and ambitious effort to apply cognitive processes to the origin of mathematical concepts. What delighted me was that in doing so, the authors helped me improve the depth of my own understanding of those concepts. I realize that many of the reviewers here and elsewhere have found errors in the presentation of the ideas, but I challenge them to offer a book that better presents those ideas in a conceptual format. Nowhere else have I read a book that describes the problems I had as a young student trying to understand the non-geometric approaches to limits and calculus. Also, their explanation of a program of discretization of continuity is one that closely resembles scientific reductionism and a similar discretization in physics. To me, finding 19 reviews here is proof enough that the book is important, accessible, and useful. The authors do seem to have a thesis that they expound past exhaustion, dealing with the metaphysics of math, but much more interesting to me is their extremely useful methodology of mapping concepts. This is something I would like to see applied to quantum mechanics, fractal geometry, set theory, and computer programming, and hope that other cognitive scientists will step up to the task. Although people who are more knowledgeable of the math literature than me may disagree, I think that this book does a scholarly job of collecting more than a few important concepts from several fields into one volume, something that is immensely helpful to persons like me at the bottom of the mathematical curve. ;)

As a physicist and recreational mathematician, I found this book stimulating and reassuring. The connection of mathematics to human realities in our embodied world gives a new way to understand the conceptual and practical power of mathematics, as well as approach its limitations. I also found it helps to explain my preference for "seat of the pants" approach to some subjects, as contrasted to the proof-driven esthetic of many professional mathematicians. I think this book may encourage new ideas in mathematics education as well. If you're a Platonist, you'll find a lot to scream about, but its a great read for any math nut.