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George Lakoff

Where Mathematics Come From: How The Embodied Mind Brings Mathematics Into Being Paperback – August 7, 2001

by George Lakoff (Author), Rafael Nuñez (Author)

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This book is about mathematical ideas, about what mathematics means-and why. Abstract ideas, for the most part, arise via conceptual metaphor-metaphorical ideas projecting from the way we function in the everyday physical world. Where Mathematics Comes From argues that conceptual metaphor plays a central role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms.

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If Barbie thinks math class is tough, what could she possibly think about math as a class of metaphorical thought? Cognitive scientists George Lakoff and Rafael Nuñez explore that theme in great depth in Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. This book is not for the faint of heart or those with an aversion to heavy abstraction--Lakoff and Nuñez pull no punches in their analysis of mathematical thinking. Their basic premise, that all of mathematics is derived from the metaphors we use to maneuver in the world around us, is easy enough to grasp, but following the reasoning requires a willingness to approach complex mathematical and linguistic concepts--a combination that is sure to alienate a fair number of readers.

Those willing to brave its rigors will find Where Mathematics Comes From rewarding and profoundly thought-provoking. The heart of the book wrestles with the important concept of infinity and tries to explain how our limited experience in a seemingly finite world can lead to such a crazy idea. The authors know their math and their cognitive theory. While those who want their abstractions to reflect the real world rather than merely the insides of their skulls will have trouble reading while rolling their eyes, most readers will take to the new conception of mathematical thinking as a satisfying, if challenging, solution. --Rob Lightner

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"A groundbreaking book." -- -Science News

"Adds body heat to the cold and beautiful abstractions of mathematics." -- -The American Scholar

"The authors take readers on a dazzling excursion without sacrificing the rigor of their exposition. Revolutionary." -- -Publishers Weekly

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Product details

Paperback: 512 pages
Publisher: Basic Books; 1st edition (August 16, 2001)
Language: English
ISBN-10: 0465037712
ISBN-13: 978-0465037711
Product Dimensions: 7.5 x 1 x 9.2 inches
Shipping Weight: 1.9 pounds (View shipping rates and policies)
Average Customer Review: 40 customer reviews
Amazon Best Sellers Rank: #744,284 in Books (See Top 100 in Books)
- #2991 in Cognitive Psychology (Books)
- #674 in Mathematics History
- #2117 in Medical Cognitive Psychology

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Antonio L. Furtado

Based on one of the four master tropes

January 2, 2019

Format: PaperbackVerified Purchase

This is a clever application of metaphor -- one of the four master tropes, discussed by Kenneth Burke in his A Grammar of Motives -- to mathematical reasoning. But the three other tropes, namely Metonymy, Synecdoche, and Irony, have also been applied to theorem proving. Irony, for example, induces proof by contradiction, where one assumes that the contrary proposition is true, and then shows that the assumption leads to an inconsistency. A new version of the book, covering all master tropes, might usefully be considered by the authors.

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Alan U. Kennington

Starts well, but runs out of steam

December 25, 2012

Format: PaperbackVerified Purchase

I bought and read this book 4 years ago. The beginning of the book makes important statements about the "true nature" of mathematics. It is the perfect counterweight to the Platonic "universe of perfect forms" which some people think explains the source of mathematical meaning. Another philosophy of mathematics is the "mathematics in the universe" idea that the universe contains mathematics, and that's where we discover it from.

In the Platonic view, mathematics is essentially completely disconnected from the physical world, but can be perceived by the mind. In the "math in the universe" view, it's "out there" outside the individual mind. The embodied mind viewpoint may be thought of as locating mathematics in the interface between individual minds and the universe around them. (Of course, there is no sharp boundary between individuals and their environments.)

The essence of this book may be summarised by a single paragraph on page 9.

"Mathematics as we know it is human mathematics, a product of the human mind. Where does mathematics come from? It comes from us! We create it, but it is not arbitrary---not a mere historically contingent social construction. What makes mathematics nonarbitrary is that it uses the basic conceptual mechanisms of the embodied human mind as it has evolved in the real world. Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history."

The rest of the book gives evidence for this claim and develops the consequences for some areas of mathematics. My favourite part of the book is the evidence for arithmetic in six-month-old human infants, using puppets, on pages 16-19. This shows clearly that some basic mathematics is not purely a matter of culture. It is innate.

Chapters 2 to 11 present a series of "metaphors" and "schemas" for various mathematical concepts --- arithmetic, algebra, logic, sets, real numbers, infinity, transfinite numbers, infinitesimals. Initially it all seems quite plausible. Integers come from adding and removing objects from collections of objects. Real numbers come from measuring sticks. But the "metaphors" get less and less credible. My marginal notes pencilled in the margins changed from very positive to slightly positive, to lukewarm, to skeptical, to strong disagreement, to utter derision. My negative marginal notes started at page 91, concerning complex numbers interpreted as rotations. At page 108, my comments are "Huh?" and "infantile" and "Why not?". On pages 109 to 111, I commented "more nonsense", "unjustified conclusion", "nonsense", "total twaddle" and "nonsense".

The metaphors are mostly like what are used to teach mathematics. In fact, a useful application of this book would be to mathematics teaching. That's what it seems like really.

I do think that this is a very good book, but the authors could have said everything much better in 150 pages instead of 493 pages. After the first couple of chapters, the rest may be considered "metaphors for teaching mathematics". The first 4 chapters are well worth reading as an antidote to the Platonic "universe of forms" philosophy of mathematics. Modern mathematics is taught as if it were "analytic" truth, i.e. as absolute, universal truth which even life-forms in other galaxies would agree with.

The fact that the basic concepts of mathematics are wired into the brain is very important. However, that is only the launch point for mathematics. The detailed content of modern mathematics cannot be mapped one-to-one with the "embodied mind".

Maybe each individual component of mathematical thinking may be identified with innate capabilities of the human mind. But all cooking uses pretty much the same set of ingredients and the same set of techniques, and yet the art of great cooks cannot be reduced to mere ingredients and techniques. Painters all use the same colours, tools, concepts of perspective and composition etc., but that does not explain all of the art in the art galleries.

The basic components of mathematical activity are indubitably in-born in the sensory/motor system of the human mind, but it is very doubtful that "cognitive science" (if it is indeed a science) can identify which mental process is being applied in each mathematical thinking-process. Probably each individual uses different innate brain functions to other individuals for the same mathematical concepts. For example, I use diagrams for almost all mathematics, whereas some mathematicians I have met say that they do all mathematics symbolically and algebraically without any visualisation.

The "embodied mind" theory also does not explain how mathematicians at some points in history totally rejected concepts which at other times were accepted as self-evident. Examples are zero, negative numbers, irrational numbers, transcendental numbers, complex numbers, infinite sets, transfinite ordinal numbers and non-Euclidean geometry. The vicious debates between intuitionists and formalists in the late 19th and early 20th centuries show that even at one point in history, there can be strong disagreement on the most fundamental ideas of what constitutes mathematics. So it seems unlikely that the "embodied mind" theory can do explain anything more than the ingredients and utensils of mathematics. I cannot explain the recipes in the cookbooks.

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Paul Hunt

Mindful Beauty

February 16, 2017

Format: PaperbackVerified Purchase

Lakoff has pursued his thesis of cognitive metaphor as embodied structure that not only let's science into the discussion of body/mind unity, but can be applied fruitfully in a variety of disciplines. Beginning by working cognitive science into linguistics research, he has expanded inquiries into literature, mathematics, and politics all with powerful effect.

I first read this book hoping it might give me some insight into how to discuss a variety of things with my brother who's study of math and physics has prevented him from imagining much value in poetry. The book marks a trail beginning with an inventory of cognitive science experiments that set the ground for basic math as cognitive metaphor, and leads to the heart warming jewel of mathematical beauty in Euler's equation. Quite the ride! When I shared it with my brother he was hooked.

Obviously, we don't all think alike, but cognitive science has begun to shed light on the basic patterns of our variety and a shared mindfulness at the heart of being human. This book is an important indication of the power of that revelation which should be shared.

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An interesting view of the nature of mathematics

May 8, 2004

Format: PaperbackVerified Purchase

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.

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