Customer Reviews

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being

5 star:		(11)
4 star:		(7)
3 star:		(2)
2 star:		(2)
1 star:		(5)

 

 

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.

I give this book 5 stars not because it is definitive and "correct," but because it proposes an exciting new tack in the philosophy of mathematics. This deeply fascinating book would have been a major addition to that philosophy, a most noble subject, were it not that the authors know little about it. For starters, they do not appreciate the extent to which intuitionists and constructivists have anticipated their attack on what they rightly deprecate as the Romance of (Platonic) Mathematics. Intuitionists entirely agree that mathematics is a human construction serving human purposes; mathematics has no existence apart from this fact.

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.

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.

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. ;)

On the whole, I enjoyed the book, and thought it was a nice overview of lots of different mathematical ideas, most of which were familiar to me, and some of which were not. As a mathematician and computer scientist with some AI/cognitive science background, I thought that some of the presentation was a little clunky. Some arguments that I find fairly easy to understand when presented as proofs were less clear when presented textually for a less mathematically inclined audience. I have three main complaints with the book.

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?

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.

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.

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.

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.

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.