Lakoff and Nunez are cognitive scientists with a deep interest in mathematics and in this book, they try to explain mathematics from a cognitive perspective. The result is fascinating. I am a mathematical realist and, as such, I have some philosophical disagreements with the authors of this book, but their explanation of the metaphors involved in some mathematical concepts I found fascinating. Furthermore, I think their ideas of embodied mathematics is fully reconcilable with an Aristotelian hylemorphic concept of reality (if not a Platonist concept of reality). Lakoff and Nunez clearly subscribe to materialism and scientism, and they conflate the brain and the mind. However, despite some clear flaws, I think all mathematics teachers and everyone interested in the ideas of mathematics should read this book.

I consider this book to be essential reading for anyone attempting to seriously understand Mathematics. In fact this book or should probably be required for anyone teaching Mathematics!I've long believed that there was no way to break down thought into discernible mechanistic-like chunks and analyze the thought process in a non-hand-waving manner. I am delighted to discover I was wrong about this. It turns out cognitive scientists have developed what seems to be a very solid method and vocabulary for doing just that, and this book explores using the methodology to analyze Mathematical ideas. The results are impressive. The authors do a good job in arguing their central notion that Mathematics can be understood as something that emerges not from an objective discovery of universal abstract truths, but from a sometimes messy set of thought processes building upon a few basic neurological capabilities, and these basic neurological capabilities are firmly embedded in our physical reality. Along the way of making this point, they do a thorough job of laying bare the conceptual trickiness in certain Mathematical ideas.The first third of the book was a little tedious. It felt like programming in Assembly language, the "close to the metal" language of computers: the operations and ideas were extremely simple and repetitive. The payoff for making it through this however was a wonderful jaunt through the land of Infinitesimals and the battles between Geometers and the Discretization program (I think a good companion to this book would be the Donald Coxeter biography King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry). The last third of the book is a case study of all the ideas inherent in understanding Euler's beautiful and somewhat mysterious special identity formula e**(pi*i)+1=0. I'm setting the book aside for now until I have more spare time to commit to consuming this last part.I come away from all this with the wish that there were a Wikipedia-like reference that gives every Mathematical idea this level of analysis. If you can speak this language of the cognitive scientist, it seems like the most clear way to express any Mathematical idea.

Like language, simple arithmetic has an instinctual basis. People can immediately see whether there is one object in front of them, two, or three, and expect that one object and one more object make two. What of more complicated mathematics? The authors argue that it is built up from simple arithmetic using metaphors, the mechanism that, as Lakoff has argued in another book, is central for cognition. For example, the conceptual jump from real to complex numbers is akin to the conceptual jump in such expressions as "time is money": time is not literally money, but there is a manner of looking at time that shows it being akin to money in some ways. Likewise, there is a manner of looking at pairs of real numbers that shows them as the real and the imaginary parts of a single complex number, akin to real numbers in some ways. Axioms are like essences in folk categories. A car usually has four wheels; there have been some three-wheel vehicles that are like cars; the definition of a car may have to change to accommodate them but not the three-wheelers that are like two-wheel motorcycles. The search is on for a definition that calls what is usually thought to be a car a car, and doesn't call what isn't. Likewise, there have been several attempts to axiomatize set theory; each either included pathological mathematical objects or excluded useful ones. The authors build up a hierarchy of mathematical objects, going all the way to nonstandard analysis, transfinite ordinals and space-filling curves, and describe the metaphors taken at each step. They take Euler's equation eπi=-1 and explain the metaphorical sense in which it is true.Frankly, I do not see the purpose of this exercise. Terence Tao proved that arithmetic progressions of prime numbers can be arbitrarily long. This proof is stored in Professor Tao's brain and in the brains of the mathematicians who have read the published proof. We do not know how it is encoded there. Although the particular encoding is specific to human biology, we do not know whether the proof itself is. Right now humans are the only beings on Earth who could understand it; yet it is very possible that within 100 years electronic computers will too. There has been an attempt to construct an automatic discoverer of mathematical concepts; although it did not progress very far, it did discover a great deal more than the overwhelming majority of students of mathematics. If mathematics is defined as the metaphorical extension of instinctual behaviors, then what it discovered is not mathematics, since a machine has no instincts. The authors mention Eugene Wigner asking why mathematics is unreasonably effective in describing the physical world and answer that it is because mathematics exists only in the brains of the physicists who are doing the describing. Yet other disciplines exist in human brains that are far less effective in describing the physical world, such as religion and philosophy; why is mathematics different?What is also lacking from the book is a sense of how unnatural mathematics is for people. Grade school arithmetic is useless except as a foundation for high school mathematics, which is useless except as a foundation for undergraduate mathematics, which is useless except as a foundation for graduate mathematics. American universities graduate about 1,200 Ph.D.s in mathematics a year, of whom half are U.S. citizens. Even if 10 times as many people are capable of mastering the program but do not, choosing other careers instead, it is still a tiny fraction of the 3-4 million babies born in the United States each year. That's not much for something that supposedly has an instinctual basis, is it?

Recently completed reading this challenging journey through Lakoff's embodied mind theory with our Philosophy of Math study group at Saint Martin's University. The group, made up of math, philosophy, and computer science professors, struggled with Lakoff's approach to how fundamentals of number, arithmetic, algebra, and infinitesimals are grounded in bodily metaphor and permutations of such metaphors through conceptual blending (for a more detailed look at conceptual blending see Fouconnier/Turner's The Way We Think: Conceptual Blending and The Mind's Hidden Complexities). There was considerable consternation with Lakoff/Nunez in that they often didn't go far enough with explanation to back up their claims and that, according to the math folks, their presentation of math concepts was sometimes plain inaccurate. Some of the group participants were clearly disturbed by the authors' full-frontal attack on Platonic mathematics and disembodied thought and chose not to attend while we read this book. Clearly many are not ready to entertain a worldview where embodiment, immanence, and emergent properties become the basis for the sacred cows of reason, math, and philosophy.Much of the trouble my colleagues had with understanding and appreciating Lakoff/Nunez's embodied mind approach was due to their lack of background with cognitive science, the philosophy of mind, consciousness studies, and of course systems and complexity theory. Their struggle highlighted to me the truism that, in today's world, specialization has a particular cost to it in that you miss out on the insights gained from contemporary approaches that are multidisciplinary or interdisciplinary such as cognitive science and systems theory. As we know from postmodern thought, we live in a pluralistic world of multiple interdependent-interacting perspectives and bodies of knowledge and to isolate oneself within a specialty or field cuts one off from the often profound insights gained from the cross-fertilization multidisciplinarity offers. My recommendation is to read other introductory books on cognitive science and embodied mind theory such as Howard Gardner's The Mind's New Science: A History Of The Cognitive Revolution or Lawrence Shapiro's Embodied Cognition to get a general sense of the context in which Lakoff and Nunez are doing their work. It also helps to read Lakoff/Johnson's original book on embodied metaphor Metaphors We Live By to get a sense of his approach.

A great read for anyone with a love for math, or with an interest in how human ideas come to be.

Perhaps this book was way above my way of thinking about certain math concepts, but I found the reading to be too dry. Especially coming from a cognitive science major. Although I may change my rating once I've re-read it outside of a term paper kind of setting, it might be a while before I would want to takle this book again.

This book was wonderful. I would reccomend it to anyone who is majoring in mathematics education, mathetmatics, or just enjoys the world of numbers.

I read Lakoff's earlier book, Metaphors we live by, and loved it. This is way more in depth. It starts strong with an introduction to what are best thought of as "hard wired" cognitive faculties like "subitizing" (instantaneous number recognition, this is also seen in other animals to a lesser extent). The authurs then build on visual schema to lay the basis for useful metaphors to comprehend higher math, at least through arithmetic and elementary logic. After that, the book goes off the rails. The embodiment idea is obscured and the authors do a poor job of making their increasingly complex case. More examples would be useful to understand the math, but they seem more concerned with showing off than being clear. The parts that were good, make the early part of the book useful so I am scoring it higher than it may deserve.

Reading this book seemed like watching a picture come in and out of focus constantly. The authors start the book with the great promise to explicitly present the underlying metaphors of all of mathematics and they begin quite well. They explain arithmetic from innate counting abilities in humans and clarifying the metaphors by which those innate abilities are extended to all of what we know as arithmetic. Unfortunately, the book starts to see-saw on the following chapter on algebra. The authors start talking about everything in terms of sets and while their explanations are lucid every once in a while, the reader – at least the reader not thoroughly familiar with such concepts as hypersets, hyperreal numbers, quaternions, etc. – start to find less and less room to grasp the concepts that the authors are trying to “clarify”. The picture comes back into focus on the chapter of the philosophy of embodied mathematics, but unfortunately the authors try to present the previously incomprehensible examples as evidence of their philosophy. In all honesty the best part of this book was the case study of Euler’s equation at the end. The only people I would recommend this to would be mathematicians, as I suspect they would have the necessary background to understand the authors’ intent.

George Lakoff, a cognitive linguist, and Rafael E. Núñez, a published their work "Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being" in 2000 as a 'cognitive science' study of how mathematics is 'embodied' and based on 'conceptual methaphors'. Cognitive science in an interdisciplinary approach towards studying how the mind works and the processes which characterize it, namely thinking. In this book, mathematics is regarded as embodied or shaped by aspects of the body. That is, the neurological or perceptual system built into the brain. However, any neural engrams involved in mathematical processing are described only at a high, abstract, hypothetical level. The book is primarily about the conceptual metaphors that underlie the foundations of mathematics. One of the most important of which the authors call the Basic Metaphor of Infinity (BMI), which is used to represent many areas of mathematics that deal of endless, unlimited sequences or processes described in numbers. I found this to be a fascinating, intriguing work, although I imagine some professional mathematicians and philosophers would still regard the substance of mathematics as existing outside of the minds of mathematicians, a view the authors reject as Platonism. George Lakoff, the better known of the two authors, would be recognized by his recent work on how conceptual metaphors can affect the narrative of political discourse. His observations on how conservatives have overtaken and undermined liberals and progressive by framing the political discussion to their advantage; e.g. the day George Bush took office "tax relief" in place of "tax burden" or "tax responsibility" began to frame how the issue would be viewed.

I was hoping for so much more from this book based upon the way it was described. At the time of reading this book, I was searching for a research-supported narrative delineating the cognitive development of mathematics as a complex web of neural networks. This book fragments the web into a series of mathematical topics and argues how the mind processes the algorithm(s) associated with that topic. There are times when pre-requisite thinking is discussed, but there is no information on how these topics bridge to others or how possibly non-mathematical topics influence the conceptualization of mathematical topics. Written by a linguist and a psychologist, there was an opportunity for the authors to discuss the formation of mathematics as a function of language and bring in cross-cultural comparisons, but nothing of this sort was part of the book.

Although I am a Mathematical Platonist. I couldn't help but be fascinated by Lakoff account of how concrete metaphors from the body and everyday experience inform our mathematical abstractions of the most aetherial and least earthy types. My only answer to Lakoff repudiation of platonism for a cognitive origin of mathematics is where does the regularity of the world (that cognitive patterns are built on) originate. An excellent Book.

I find some of the arguments in this book tautological, thought it is difficult to articulate why. The section on an Embodied Philosophy of Mathematics is one of the most interesting in the book. The authors argue against "The Romance of Mathematics" (Platonism, plus some cultural effects) and against Postmodernism as a philosophy of mathematics. Their solution to "what is mathematics" lies somewhere in the middle: every human has certain basic cognitive capabilities. Based on these capabilities, we create metaphors; from these metaphors we build up mathematics. Therefore math is not totally arbitrary, because it is based on universal human neurological characteristics; neither is it universal or somewhere "out there", because it is created and practiced by humans. This philosophy allows room for both cultural and historical contingency and universality. The writing is very clear, but readers with little or no recollection of calculus might find parts of it difficult.

Really provoking and intereasting. An evolutionary and Psychobiological essay on emergence of Mathematical basic entities.

A superbly written mathematics book for geeks and non-geeks alike. OK it's better if you're a little geeky. The author is a linquist, and provides compelling metaphorical explanations for difficult concepts. The Appendix contains a lucid explanation of the famous Euler 'Magic equation'. That alone is worth the price of the book. I honestly haven't read this book in its entirety, it's not that kind of book. I just keep going back to it again and again, based on my interest of the moment.

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