Rating: 
Summary: Basic Premise Is Flawed
Review: ( This is a corrected version of the review I previously submitted.)
Firstly, I must admit I have not read the book. I base my review on the editorial review and customer reviews which reveal enough about the book to justify my conclusion that the Book's premise is flawed. Secondly, I should state I have great admiration for Lakoff who is one of the pioneers in metaphor dynamics and its being recognized an essential area of study in cognition.
However, the Book, "Where Mathematics Comes From" is flawed in that it attempts to describe the very mathematics that determines who we are and how our metaphorical conceptions evolve. We live in a Universe whose processes are determined by mathematical and physical laws. As such our embodied metaphorically predisposed minds are very much a product of that mathematics. Turner and Nunez have erred on this point. Our metaphors may determine how we perceive mathematics but, more importantly, it is the mathematics that determines how we become predisposed to use metaphor. As such, it is not mere fortuitousness that allow us to fairly accurately perceive the mathematics governing the Universe. We had to make an accurate assessment based upon the mathematics determined by the probability that it would improve the chances of our survival as a species. In other words, we had to replace the mathematical intuitions stemming from metaphors that were wrong such as the pre-Galileo geocentric theories to those that were correct or more correct and, sure enough, place a greater emphasis on the scientific method--our species' survival depended upon it. I think, Lakoff and Nunez have merely restated Darwinianism in metaphorical terms.
Rating:
Summary: Basic Premise Is Flawed
Review: ( This is a corrected version of the review I previously submitted.)
Firstly, I must admit I have not read the book. I base my review on the editorial review and customer reviews which reveal enough about the book to justify my conclusion that the Book's premise is flawed. Secondly, I should state I have great admiration for Lakoff who is one of the pioneers in metaphor dynamics and its being recognized an essential area of study in cognition.
However, the Book, "Where Mathematics Comes From" is flawed in that it attempts to describe the very mathematics that determines who we are and how our metaphorical conceptions evolve. We live in a Universe whose processes are determined by mathematical and physical laws. As such our embodied metaphorically predisposed minds are very much a product of that mathematics. Turner and Nunez have erred on this point. Our metaphors may determine how we perceive mathematics but, more importantly, it is the mathematics that determines how we become predisposed to use metaphor. As such, it is not mere fortuitousness that allow us to fairly accurately perceive the mathematics governing the Universe. We had to make an accurate assessment based upon the mathematics determined by the probability that it would improve the chances of our survival as a species. In other words, we had to replace the mathematical intuitions stemming from metaphors that were wrong such as the pre-Galileo geocentric theories to those that were correct or more correct and, sure enough, place a greater emphasis on the scientific method--our species' survival depended upon it. I think, Lakoff and Nunez have merely restated Darwinianism in metaphorical terms.
Rating: 
Summary: The Enlightenment Continues...
Review: A vert nice sequel to Philosophy in the Flesh. In nails down this method of philosophy as undeniably valid and noteworthy. For anyone interested in how it is that something is known about the world, these two books together are critical. It may take a long time before the epistemological basis of bodily experience becomes mainstream, but if you want to understand how it is you understand, then you can't do better than starting here. The collective contributions of the ladies and gentlemen who have conducted the research necessary to this effort cannot be overestimated. I am so grateful to have these references.
Rating: 
Summary: What about Brouwer's Intuitionism?
Review: Anyone who claims that this book is profoundly new or refreshing is ignorant of past achievements of L.E.J. Brouwer and his program of Mathematical Intuitionism, which stated that mathematics is solely a human cognitive activity based on a mathematician's intuition of the continuum. Intuitionism went on to claim that mathematical formalisms (symbols, languages, etc) are not mathematics, and the Platonic idealism is something short of religious - not belonging in mathematics.What is worse is how the authors themselves seem to have overlooked Brouwer's Intuitionism too, giving the impression that in some ways; they were "pulling a Christopher Columbus" or "reinventing the wheel" - discovering the discovered and creating the created. However, this is only partially true. While the authors make many claims that are similar to Brouwer's Intuitionism, there approach is completely different. The authors attempt to use science to describe mathematics, while Brouwer used self-exploration, allowing him to stay within the validity and certainty of solipsism. Note that Brouwer would most likely consider the authors' approach completely baseless or possibly even hypocritical, considering his strong solipsist stance on the disability of science to portray the Truth... but that is another topic for many other days. Most people choose to guess their way out of solipsism, and so we have the sciences. However, strong skepticsm should always be paired with the claims of science. The scientific approach used in this book, to describe "where mathematics comes from", is interesting, to say the least, and inspiring to say the most. In fact, I think that it is a very important area of research, and I highly recommend the book, even though I disappointed with the author's lack of attention to Brouwer's Intuitionism. Finally, it is my opinion that science shouldn't be used as a foundation for mathematics, but I still think that it can be used as a source of inspiration for a better foundation of mathematics - that is why I recommend this book. (By the way, the book is also, simply allot of fun to read... well, I am interested in foundations of math, so maybe I am biased.)
Rating:
Summary: We need more books like this one in other fields
Review: 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. ;)
Rating:
Summary: Refreshing approach to the ideas of mathematics
Review: 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.
Rating:
Summary: Basic Premise Is Flawed
Review: Firstly, I must admit I have not read the book. I base my review on the editorial review and customer reviews which reveal enough about the book to justify my conclusion that the Book's premise is flawed. Secondly, I should state I have great admiration for Lakoff who is one of the pioneers in metaphor dynamics and its being recognized an essential area of study in cognition.
However, the Book, "Where Mathematics Comes From" is flawed in that it attempts to describe the very mathematics that determines who we are and how our metaphorical conceptions evolve. We live in a Universe whose processes are determined by mathematical and physical laws. As such our embodied metaphorically predisposed minds are very much a product of that mathematics. Turner and Nunez have erred on this point. Our metaphors may determine how we perceive mathematics but, more importantly, it is the mathematics that determines how we become predisposed to use metaphor. As such, it is not mere fortuitousness that allow us to fairly accurately perceive the mathematics governing the Universe. We had to make an accurate assessment based upon the mathematics determined by the probability that it would improve the chances of our survival as a species. In other words, we had to replace the mathematical intuitions stemming from metaphors that were wrong such as the heliocentric theories to those that were correct or more correct and, sure enough, place a greater emphasis on the scientific method--our species' survival depended upon it. I think, Lakoff and Nunez have merely restated Darwinianism in metaphorical terms.
Rating:
Summary: An interesting view of the nature of mathematics
Review: For as long as Western mathematics has been around, it has generally been viewed as having an existence independent of human experience, as belonging to a Platonic realm of forms and ideas. To make it embodied in the human psyche, as the authors attempt to do in this book, would be a sacrilege to many mathematicians. Such a move would deny the 'eternal truth' of mathematics some would argue. But the last few decades have seen the rise of cognitive science, and this field has led to many interesting insights into the operation of mind and has demystified its status in the world. The authors though see cognitive science as being deficient in one respect: it has omitted the study of mathematical ideas from a cognitive perspective. There is no cognitive science of mathematics, they say, and hence they endeavor in the book to correct this deficiency. Such a project is definitely worth the effort, for mathematics has to be interpreted in the light of what is known about the mind, or as the authors put it, "it should study precise nature of clear mathematical intuitions". The book is very interesting to read, and the justifications for the assertions put forward by the authors are certainly the most optimal if viewed in the context of what is currently known in cognitive science. Further work must be done however, particularly in tying their ideas to the very intensive research in neuroscience that is being done at the present time. The prospect of having a science of mathematical thought is an exciting one. This book is the best that is currently available. The attitude of the authors is most refreshing, in that they not only show great enthusiasm throughout the book, but they are not nervous about discarding what they view as the "romance" of mathematics. They list several statements illustrating this "beautiful romance", such as the view that mathematics has an objective existence, which transcends the existence of human beings; or that human mathematics is merely a part of abstract, transcendent mathematics, and that reason is a form of mathematics. These romantic beliefs appear to be false, the authors say. Instead, they argue, the nature of mathematical ideas is that they are inherently metaphorical in nature. They give several examples of this in the first few pages of the book, with the rest of the book elaborating in great detail their reasons for asserting this. This is certainly an exciting time to be involved in mathematics, and assuming more evidence is accumulated that supports the authors opinions on the embodied nature of mathematics, it will be even more interesting to be engaged in mathematical research and in the teaching of mathematics. Mathematical thinking will then viewed as part of us, not some abstract collection of statements existing in some vaguely defined realm. Viewing mathematics as purely embodied may also give much more insight into teaching non-human machines how to do mathematics. This is the most exciting prospect of all.
Rating:
Summary: The endorsement of a high school Calculus teacher
Review: 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 created a situation where math is feared (like a god). An earlier reviewer took issue with this point by stating that pi is pi no matter where one goes. I hope I have illustrated from my opening paragraph how we may have miscalculated how essentially human pi really is (and remains). Pi is *our* pi! Even given this point, Lackoff and Nunez still affirm that math works predictably. Our observations of the world of circles, no matter where we go, should predictably conform to our infinite number called pi (depending on the accuracy of the circle). But pi is still our number (and that's the point). Experience tells us that circles approximate pi, so we are wise to have it as it is. But that does not mean pi is outside of us, or anything other than something the human brain came to via neural pathways. There is no necessity of postmodernism in so arguing, and the authors point this out. This does not open the door to relativistic math.
2) Not all thoughts are conscious thoughts. This might help explain why we can speak so quickly while conforming to standard grammar (without thinking about conjugations, tense agreements, etc.). We think at a level we don't consciously know about, yet thinking is more than what we know about. That is, our brain is more complex than what is at the front of our mind. Perhaps you recall a "eureka" moment that came suddenly, as if from regions below the conscious radar.
3) Metaphor is how we know things, and the mapping of one domain of thought to another domain (for the sake of understanding) is metaphor. It is metaphor because of essential realities, and not mere simile. That is, metaphor is more than a device of literature -- it is a way of grasping, and it involves the transfer of ideas between realms; it works as the basis of conceptualization. We know our subject in the abstract (if we really know it) by having some connections with domains of discourse already familiar to us. We know by analogy but not merely because the analogy is handy, but because the analogy holds in a fundamental way. For an exercise, imagine how Venn Diagrams correspond to the simple idea of containment -- perhaps transfer your understanding of the Venn Diagram back onto real pottery containers. Now consider how we abstract Boolean logic beyond the pottery -- even beyond the metaphor of containment and Venn Diagrams -- into pure symbolic language. After enough layers of abstraction are piled on, the original conceptual grid seems forgotten, and so gives the life-support to mystical notions of math (hence the Romance that the symbolic world of math is only accessible to the enlightened). But math is not abstract; it starts from human experience!
As soon as I started reading this book, I found many new and useful ways to communicate math to my students -- both in ways conceptually satisfying to them and honest to the math.
One strategy often used to make math easier to students is to go over the history of the math under discussion. But let's be honest, knowing the history of Euler is not going to automatically reveal the structure of his mind (a structure that made his math obvious to him). This book takes us beyond the history of math ideas, to the structure of math ideas. This book takes us to the concepts that make the ideas accessible to the student's human mind. By knowing the elementary concepts behind e, for example, we can gain more mileage than knowing its historical circumstances. And so the book ends with a case study of e, and this alone should help a teacher who needs to reach the human mind of the student.
This is the math book that I have been looking for as a teacher. It is the math book I have been looking for after having read Russell and his philosophy of math. This book is the book I was looking for in order to understand how metaphor works in human history. It has application well beyond math, and its application to math is beyond anything I normally encounter. I cannot give this book high enough rating.
Rating:
Summary: One further clarification -- on Q as an ordered field.
Review: I have given 5 stars, so I don't adversely affect the book's rating. However, there was one further clarification which I wanted to add. A large portion of the book concerns itself with the extension of the rational number system to the real number system, as first developed rigorously by Dedekind and Cantor. A big deal is made that the "gaps" found in the rational number system are only apparent, i.e. that the "gaps" only exist when one considers the rational numbers either in relation to the real numbers, or "metaphorically" mapped onto some other "line", (the "ordinary, usual line", for instance.) Lots of quotes from Dedekind are used, when he explains these "gaps" in geometric language in terms of a line. The authors imply that because of this, it's incorrect to conclude that the gaps necessarily exist as a consequence of the rational numbers themselves, and that the gaps only exist when one considers the "line". However, this is false. The "gaps" found in the rational number system are a fundamental defect of the rational numbers, and have nothing whatsoever to do with the real numbers, complex numbers, or any other line or geometric metaphor. A picture is presented in the book of the set of rational numbers, enclosed in a circle, and pointing to it is the quotation: "Look, no gaps!", implying that by themselves, the rational numbers have no gaps. This shows a misunderstanding of the nature of the gaps, however. To say that the rational numbers have "gaps" is to say that as an ordered field, the rational number system is not complete, i.e. NOT every set of rational numbers bounded above has a least upper bound. It is possible to look ONLY at the rational number system, and yet construct a perfectly legitimate set of non-empty rational numbers, bounded above, which has NO least upper bound. It is this least-upper-bound property which is missing in the rationals, and no reference to the reals is needed to point this out. If one claims the rationals have no "gaps", then one must be talking about some other property of the rational numbers, besides its properties as an ordered field. But if this is the case, why are you writing about Dedekind cuts and Cantor sequences in the first place?
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