Rating: 
Summary: you must read this book
Review: If you hate math, you must read this book.
If you love math, you must read this book.
If you live in the deadly icy Platonic realms, you must save your life by reading this book.
If your math teacher is confusing the hell out of you, get this book for your math teacher.
Rating: 
Summary: Good ideas, but self-aggrandising and full of blunders
Review: Lakoff and Nuñez have a very interesting general framework for approaching their topic: mathematics arises by the extension of innate human capacities (e.g. subitization) or basic universals of human experience (spatial and motor experience), and the means of extension is cognitive metaphors which preserve the basic inferential structure of the source domain. The first few chapters provide a plausible sounding, perhaps workable account of arithmetic, simple logic and set theory, but one that they should have developed in far more detail (e.g. their account of intersection in terms of container schemas is criminally underexplained).The biggest problem is that they do not stop there. They proceed from the speculative but plausible ideas I just mentioned into butchering higher mathematics in ways that have been amply documented in reviews both here and elsewhere. (To add an example, they massacre the Compactness Theorem for logic, by absolutely failing to mention the fact that it only works for first-order logic. Their "account" of "a mathematician's understanding" of the Theorem is just wrong, since you can't really be said to understand it if you don't know why it fails for second-order and higher logic, much less if you don't know the fact in the first place). More worrrying is their completely misinformed philosophical attack on mathematical realism. They manage the task of listing in their bibliography more than one fundamental collection on philosophy of math, yet completely ignoring their contents. They attack a "folk Platonism" that, no matter how popular it may be among actual mathematicians, is actually widely criticised in the philosophy of mathematics (the case of Brouwer, that somebody below mentions, is only one of many). They pass off as their own well-known arguments (e.g. a version of their argument against numbers as abstract objects, that we can't decide among the numerous candidates, was given by Benacerraf I believe in the late 60s, and it could well predate that). If they want to argue the philosophy of math they should read philosophers of math and engage them. Otherwise it is extremely hard to take them seriously. In short, they have very interesting ideas, but the mass of technical vagueness and blunders, plus the big strawman that is their "philosophical" argument, suggests that they are more interested in passing off as intellectual revolutionaries among the pop-science book audience than in contributing to our understanding of the topic.
Rating: 
Summary: Don't be too critical
Review: One star to balance the 5 back to the original 3 I had ORIGINALLY given. I'm the "Q as an ordered field" guy. Apparently, I didn't actually check, but my ORIGINAL long, highly critical (but less than 1,000 word) review was rejected or for some reason not displayed. I guess you can't be too critical. Well, I don't have time to reconstruct my review, so here's the gist: The authors say their book is cognitive science and should not be taken as mathematics. However, they use so much terminology, vocabulary, symbolic manipulation and arguments from mathematics that a lot of their book seems indistinguishable from math. Even worse, throughout the book they seem to be attacking MATHEMATICAL statements using cognitive science, when they themselves said math was not their objective. Not consistent at all.
Rating:
Summary: The first serious study of the cognitive science
Review: The only math ideas humans can have are ideas invented by the human brain: Where Mathematics Comes From argues that a conceptual metaphor plays a defining role in math ideas within the unconscious, drawing important connections between mathematical ideas and what they mean. It's the first serious study of the cognitive science of where math ideas originate: college-level audiences will find it involving.
Rating: 
Summary: classic lakoff
Review: this book is a linguist's assessment of the origin of our cognitive mathematical faculties. it is a good read, and is more satisfying than most pop-sci type books. I highly recommend reading George Gamow's 1,2,3...Infinity! along with this book.
Rating:
Summary: classic lakoff
Review: this book is a linguist's assessment of the origin of our cognitive mathematical faculties. it is a good read, and is more satisfying than most pop-sci type books. I highly recommend reading George Gamow's 1,2,3...Infinity! along with this book.
Rating: 
Summary: Pretty good...if you can deal with a few things...
Review: This book is well written and has some interesting ideas. But, to be honest, I couldn't finish it. The authors make too many errors in their arguments. Their proof that pi is not real annoyed me, it was like listening to someone spouting psychobabble and realizing that they're not saying anything at all. The authors seem to have difficulty noting the difference between the way something is visualized/explained and the way it actually is. Another thing that bothered me is that their central thesis (the mind creates math) is based on the idea that "the only math people can understand is the math that the brain can handle." That statement is vacuously true and does not show (as they want) that there does not exist math that the brain cannot handle. I'm really going off on a tangent, so I'll stop here. But if you can ignore things like that, this is a very interesting book. I'm probably just being too hard on it since it attacks some of my beliefs (sometimes very shoddily, I'd like to think).
Rating:
Summary: at least it will get you thinking. . .
Review: this is one of two books chosen for my summer study in the philosophy of mathematics, and i must say, it is a gigantic let down. many of the critiques of this book are based on the mathematics being wrong(you can go to nunez's web site to get links for those reviews), but there are simple mistakes in arguments made throughout the book(particularly when they consider the cardinality of infinite sets). the authors not only make mistakes late in the book, they start in right away with mistaking algorithms and definitions. the book never even approches number theory, topology, or geometry, and their critiques of algebra and calculus do not carry over. lastly, i must point out, the book reads like a last minute english b.s. paper. they seem to have gone to great lengths to fill space by repeating things time and again and let's not even get started on their endless columns. if you want writings that are supposedly on how math is created(whether you believe that or not), they i would recommend you look for a book by mathematicians who are not as likely to butcher the very target of their inquiry.
Rating:
Summary: Traces back all math to simplest observations. Long read.
Review: Whenever a person finds out that I'm a math enthusiast, 9 times out of 10 I get an uncomfortable reaction along the lines of "Oh, I HATE math!" In my experience, the mathphobe's biggest gripe is that math is a completely abstract concept, all based on memorization of some strange language, with so much of it having absolutely no comparison to the physical world. This book strives to show that mathematics, from basic arithmetic to more advanced branches, can in fact all be reduced down to mental metaphors of physical concepts. Early in the book, the authors present the sound scientific evidence that humans have an innate understanding of the concept of quantity, and some degree of manipluation with quantity. This ultimately leads to an understanding of addition, and then subtraction. Those concepts, combined with the understanding of how to group objects in like sets, leads to an understanding of multiplication (add like sets) and division (subtract like sets). The book then introduces a few more fundamental ideas that the human brain can use to make analogies with (motion along a path, rotation, etc.), and recreates more common mathematical concepts in increasing complexity: geometry, trigonometry, logic, set theory, etc. At the end the book the authors even successfullly tackles Euler's equation (e^i*pi = -1), a classic example of something in mathematics that doesn't make any logical sense at first glance. The book is extremely thorough in the way it presents all this. Most chapters start off by introducing a new cognative metaphor, then including a table showing the mathematical concepts to be presented and to which cognative metaphor each one relates. For a book on mathematics, this is actually a rather long read. It's thorough because it has to be, given the subject and the authors' claims. But the book might seem to drag around the middle, with a lot of repitition in each chapter as the strategy in breaking down the mathematics is constantly applied. Still, I found this to be an overall very interesting read. I think the authors succeed in showing how all sorts of math concepts break down to the simplest fundamentals, which in turn can be mentally assocated with concepts we can understand in the real world.
Rating:
Summary: Traces back all math to simplest observations. Long read.
Review: Whenever a person finds out that I'm a math enthusiast, 9 times out of 10 I get an uncomfortable reaction along the lines of "Oh, I HATE math!" In my experience, the mathphobe's biggest gripe is that math is a completely abstract concept, all based on memorization of some strange language, with so much of it having absolutely no comparison to the physical world. This book strives to show that mathematics, from basic arithmetic to more advanced branches, can in fact all be reduced down to mental metaphors of physical concepts. Early in the book, the authors present the sound scientific evidence that humans have an innate understanding of the concept of quantity, and some degree of manipluation with quantity. This ultimately leads to an understanding of addition, and then subtraction. Those concepts, combined with the understanding of how to group objects in like sets, leads to an understanding of multiplication (add like sets) and division (subtract like sets). The book then introduces a few more fundamental ideas that the human brain can use to make analogies with (motion along a path, rotation, etc.), and recreates more common mathematical concepts in increasing complexity: geometry, trigonometry, logic, set theory, etc. At the end the book the authors even successfullly tackles Euler's equation (e^i*pi = -1), a classic example of something in mathematics that doesn't make any logical sense at first glance. The book is extremely thorough in the way it presents all this. Most chapters start off by introducing a new cognative metaphor, then including a table showing the mathematical concepts to be presented and to which cognative metaphor each one relates. For a book on mathematics, this is actually a rather long read. It's thorough because it has to be, given the subject and the authors' claims. But the book might seem to drag around the middle, with a lot of repitition in each chapter as the strategy in breaking down the mathematics is constantly applied. Still, I found this to be an overall very interesting read. I think the authors succeed in showing how all sorts of math concepts break down to the simplest fundamentals, which in turn can be mentally assocated with concepts we can understand in the real world.
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