Where Mathematics Come from: How the Embodied Mind Brings Mathematics into Being Paperback – 26 Jul 2001
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About the Author
George Lakoff is Professor of Linguistics at the University of California, Berkeley. He was a founder of the generative semantics movements in linguistics in the 1960s and of the field of cognitive linguistics in the 1970s, and one of the developers of the neural theory of language in the 1980s and '90s. He is the co-author, with Mark Johnson, of Metaphors We Live By and Philosophy in the Flesh.Rafael Nunez is currently at the Department of Psychology of the University of Freiburg, and is a research associate of the University of California, Berkeley. He is the co-editor of Reclaiming Cognition: The Primacy of Action, Intention and Emotion.
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Top Customer Reviews
Some elements of the authors' approach were new to me (a science student in the 1960s), and they offer a more practical and far less abstract method of considering the higher forms of maths. If you have no mathematical background at all, then it is definitely not for you; but if you spent several years enjoying the struggle with maths at school and mostly succeeding with a brilliant teacher, but then getting lost at Uni because of deadly dull soporific lecturers, then it could well be of benefit.
It is still hard work, and I was tempted to give up several times over a couple of months. But something kept drawing me back to it, and I managed to find the time to work on it sufficiently to appreciate why I previously had problems, and why with a slightly different approach it could all have been so much easier. I'm still no expert, but some of the mysteries have been revealed. Now, if only they could apply the same lucid reasoning to Quantum Mechanics...
Most Helpful Customer Reviews on Amazon.com (beta)
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.
The philosophy of math has attracted some fine and exciting minds since Frege published his Begriffschrifft in 1879. Around 1900: Russell, of course, but also Husserl. Around 1940: Godel, Quine, Fraenkel, Bernays, Church, Curry, Brouwer, Weyl. More recently: Chihara, Boolos, Parsons, Resnick, Maddy, Shapiro, Detlefsen, Hartrey Field, Burgess, Rosen, Putnam. Regrettably, Lakoff and Nunez appear to have assimilated almost none of this literature.
This cognitive business will eventually have to interact with logic and Ed Zalta's formal theory of abstract objects.
Mathematicians reviewing this book in professional journals, as well as a number of Amazon reviewers, have complained of serious failings in L&N's understanding of some mathematical points. Lakoff argues that the errors found in earlier printings of WMCF are now corrected. On verra...
Readers should keep in mind that Lakoff is a linguist who made his reputation by linking linguistics to cognitive science and the analysis of metaphor. Nunez is a product of the Swiss school of mathematics as grist for cognitive psychology, founded by Piaget. Neither is well-trained in logic, the philosophy of set theory, the axiomatic method, metamathematics, nonstandard analysis and the ontological presuppositions of calculus, the derivations of number systems, and so on.
This book builds on two earlier fine books by Lakoff, his (1987) "Women, Fire and Dangerous Things" and his (1999 with Mark Johnson) "Philosophy in the Flesh.". Both books are very far from academic writing at its worst, but their probing analyses of metaphor, Image Schemata, and other concepts from second-generation cognitive science are not easy. Lakoff (1987) was fascinated by some technical ideas of Putnam's, about which WMCF is unaccountably silent. Lakoff and Johnson contains philosophical riches (thanks to Johnson, a significant contemporary philosopher) that I miss in WMCF. Lakoff and Nunez rightly invoke the authority of Saunders MacLane in support of their position. The authors acknowledge Reuben Hersh very warmly, but do not seem acquainted with his (with Philip Davis) "The Mathematical Experience." WMCF does not cite J R Lucas's "The Conceptual Roots of Mathematics" at all.
Nunez has devoted much of his career to thinking about the foundations of analysis, the real and complex numbers, and about what he calls the Basic Metaphor of Infinity. These topics, worthy though they be, form part of the superstructure of mathematics. The efforts of cognitive science should, I submit, be redirected to the foundations thereof. Now Lakoff and Nunez do pay a fair bit of attention early on to logic, Boolean algebra, and the Zermelo-Fraenkel axioms. And they do linger a bit over group theory. But logic, set theory, number systems, algebra, relations, mereology, topology, and geometry, more or less in that order, should have been the primary focus of their investigation.
I sense that many working mathematicians resist the approach and conclusions of Lakoff and Nunez. This situation is to be regretted. Mathematics has become an extremely powerful toolbox for the mind. Logic and abstract algebra have much to offer to the social sciences and humanities. But communicating these riches to the wider community has proved difficult, and the problem is worsening. For instance, it seems that set theory has vanished from the school curriculum. My students tell me they do not even hear the word "set" spoken until their second year at university. (I learned the core of intuitive set theory around age 12 in the 1960s, and the power of set theoretic metaphors has delighted me ever since.) Even something as basic as first order logic is nowadays learned only by the more technical philosophy majors, and by a small subset of math majors. Hence only a few specialists learn more than calculus, applied statistics, differential equations, and a bit of linear algebra. Just how many persons with a university education know what an equivalence class is? A partial order? A morphism? What it means for a set of axioms to have a model? It is my hope that the cognitive approach to mathematics will suggest improvements to the toolbox of abstractions, and better ways to communicate that toolbox to nonspecialists.
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. ;)
The first is with their technique of "mathematical idea analysis", in which they state that a particular metaphor is being applied in some area of mathematics (between two mathematical domains or between a mathematical domain and some conception of the real world), and then provide an explicit mapping between concepts in the two domains. I think the concept is great, but after a few examples it became fairly tedious, and seemed like filler. Maybe this wouldn't be a problem for someone who was less familiar with the domains under discussion.
The second is that while the book did a great job of describing the metaphors and conceptual mappings, it didn't do such a good job of providing evidence that people are actually using these metaphors when doing these kinds of math. Suppose I claim that when people do modulo 3 arithmetic, they are really using mental mechanisms evolved to deal with traffic lights. Even if you think it's a good metaphor (which it probably isn't, for several reasons), it's certainly not obvious a priori that it really describes what's going on cognitively. There may well be experiments to test t he hypothesis, but they would have to be very careful not to confuse correlation and causation. Although I'm confident that Lakoff and Núñez are doing experiments to back up their claims, I don't think the book discussed such experiments sufficiently.
My third complaint is that the book seems to suggest that if a mathematical idea is not obvious or "inherent", it must be a metaphor. It is not obvious that the earth orbits the sun. Does this mean that when we think about the earth orbiting the sun, we are necessarily doing it metaphorically? In particular, I feel that the book treats zero and the empty set or collection unfairly. Just because it took people a while to start using them does not mean that they only exist as the products of metaphor. The authors seem to have a particular problem with the empty collection, and especially confuse it with the absence of a collection, even in the concrete domain of physical objects. I think the problem is that when they talk about a collection of objects in space, they do it in the absence of a notion of boundaries or containers. I can see how a person might have trouble distinguishing an empty collection of objects from the absence of a collection in a vacuum, but who has trouble distinguishing an empty bag of Scrabble tiles from not having a bag of Scrabble tiles, or a circle of string with no marbles inside from an empty floor with no circle of string?
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