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

George Lakoff & Rafael E. Núñez


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Reviews


ABC News, by John Allen Paulos
Journal for Research in Mathematical Education, by Norma Presmeg
Politiken (Denmark) by Ib Ravn [in Danish]
The American Scholar
, by John Allen Paulos
Publisher's Weekly
Science News
Der Bund, by Daniel Goldstein [in German]
Mathematical Association of America (MAA), by Bonnie Gold
Author's reply to MAA's review
Santa Fe Institute - Business Network
Dallas Morning News
Internet Book Watch
Amazon.com
Newcity Chicago, by Nathan Matteson
C&RL News
American Library Association, by Bryce Christensen
American Scientist
From the Back Cover
Village Voice (VLS) Feb 2001 Bestseller list

 

From Publisher's Weekly®  

This groundbreaking exploration by linguist Lakoff (co-author, with Mark Johnson, of Metaphors We Live By) and psychologist Núñez (co-editor of Reclaiming Cognition) brings two decades of insights from cognitive science to bear on the nature of human mathematical thought, beginning with the basic, pre-verbal ability to do simple arithmetic on quantities of four or less, and encompassing set theory, multiple forms of infinity and the demystification of more enigmatic mathematical truths. Their purpose is to begin laying the foundations for a truly scientific understanding of human mathematical thought, grounded in processes common to all human cognition. They find that four distinct but related processes metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick and moving along a path. By carefully unfolding these primitive examples and then building upon them, the authors take readers on a dazzling excursion without sacrificing the rigor of their exposition. Lakoff and Núñez directly challenge the most cherished myths about the nature of mathematical truth, offering instead a fresh, profound, empirically grounded insight into the meaning of mathematical ideas. This revolutionary account is bound to garner major attention in the scientific press--but it remains a very challenging read that lends itself mostly to those with a strong interest in either math or cognitive science. (Nov. 15) Copyright 2000 Cahners Business Information.

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From Santa Fe Institute - Business Network

Cognitive Structure of Mathematics.

Rafael Núñez and George Lakoff have been able to give an elaborate first answer to the questions: How can advanced mathematics arise from the physical brain and body? Given the very limited mathematical capacity of human brains at birth, how can advanced mathematical ideas be built up using the basic mechanisms of conceptual structure: image-schemas, frames, metaphors, and conceptual blends?

In particular, have been able to solve three difficult cases: (1) the grounding of arithmetic, set theory and formal logic in the brain and body. (2) The cognitive structure of actual infinity (infinity as a "thing") for a wide variety of cases: the infinite set of natural numbers, points at infinity, mathematic induction, infinite decimals and the reals, limits and least upper bounds, infinitesimals and the hypereals, and tranfinite numbers. (3) The conceptual structure characterizing the meaning of e-to-the-power-ix, allowing us to characterize in cognitive terms what Euler's equation e-to-the-power-[pi]i + 1 = 0 actually means and why it is true on the basis of what it means. The result is an extended start on an embodied theory of mathematical ideas growing out of, and consistent with, contemporary cognitive science.

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From Internet Book Watch

Where Mathematics Comes From represents a unique collaboration between linguist Lakoff and psychologist Nunez as they study ideas and how mathematics has evolved. Links between consciousness, math, metaphor and daily life provide new and important details on the systems of math and how they arise from the brain. An excellent, detailed survey.

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From Amazon.com

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 --This text refers to the Paperback edition.

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From Dallas Morning News

Math may be not in the stars, but in ourselves.

Some people believe in magic. Some believe in mathematics. And some people believe that math is magical. Somehow math manages to describe the world with uncanny accuracy. Scientists use math to calculate the timing of eclipses, the power of nuclear explosions, and the trajectories for sending spaceships to the moon.
Math is essential for everything from designing jets to building bridges. Math works better than anything else in the world, and scientists have often wondered why. Four decades ago, the physicist Eugene Wigner expressed that wonder in a famous essay, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." "The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it," Dr. Wigner wrote. Many scientists suspect that math's success signifies something deep and true about the universe, disclosing an inherent mathematical structure that rules the cosmos, or at least makes it comprehensible. Somehow humans have been able to discover the laws governing reality in the form of symbols and the rules for combining them. Or so it seems.

But not to George Lakoff and Rafael Nunez. As cognitive scientists, they see a world ruled not by math, but by the human brain. Whatever might be "real," they write, human knowledge depends solely on the brain and its own ways of finding things out. Math is not a discovery about the external world, but an invention rooted in metaphors linked to human thoughts, sensations and actions. "Where does mathematics come from?" Drs. Lakoff and Nunez ask. "It comes from us," they answer in their new book, Where Mathematics Comes From (Basic Books). "We create it, but it is not arbitrary," they write. "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." Drs. Lakoff (of the University of California, Berkeley) and Nunez (of Berkeley and the University of Freiburg in Germany) contend that all mathematical ideas are elaborate metaphors. Those metaphors are drawn from real-word experience and then linked and mixed to guide the various realms of mathematical practice. Basic principles of arithmetic, algebra, trigonometry, mathematical logic and other math subdivisions all rely on metaphorical reasoning to generate rigorous systems of deduction and calculation. Arithmetic, for example, can be envisioned as movement along a path with markers placed at equal intervals, the metaphorical basis for the concept of a number line. Other metaphors can be identified to illustrate the ideas underlying more advanced math. Therefore, Drs. Lakoff and Nunez conclude, math is a mere human invention, a systematic way of capturing the way the brain sees the world. "The only mathematics that we know is the mathematics that our brain allows us to know," Dr. Lakoff said in San Francisco last month at a meeting of the American Association for the Advancement of Science. Consequently, he says, any question of math's being inherent in physical reality is moot, since there is no way to know whether or not it is. "Mathematics may or may not be out there in the world, but there's no way that we scientifically could possibly tell," Dr. Lakoff claims. Math succeeds in science, Drs. Lakoff and Nunez argue in their book, only because scientists force it to. "All the 'fitting' between mathematics and the regularities of the physical world is done within the minds of physicists who comprehend both," they write. "The mathematics is in the mind of the mathematically trained observer, not in the regularities of the physical universe."

All this is very interesting, and has earned their book a reasonably high Amazon.com sales ranking. But their analysis leaves at least a couple of questions insufficiently answered. For one thing, the authors ignore the fact that brains not only observe nature, but also are part of nature. Perhaps the math that brains invent takes the form it does because math had a hand in forming the brains in the first place (through the operation of natural laws in constraining the evolution of life). Furthermore, it's one thing to fit equations to aspects of reality that are already known. It's something else for that math to tell of phenomena never previously suspected. When Paul Dirac's equations describing electrons produced more than one solution, he surmised that nature must possess other particles, now known as antimatter. But scientists did not discover such particles until after Dirac's math told him they must exist. If math is a human invention, nature seems to know what was going to be invented.

Still, Drs. Lakoff and Nunez make a strong case that people have built mathematics out of concepts drawn from human experience. Yet somehow that realization does not resolve the old mystery about why math works so well, but rather deepens it.

By Tom Siegfried (tsiegfried@dallasnews.com).
03/05/2001, The Dallas Morning News. Copyright 2001 Gale Group Inc. All rights reserved. COPYRIGHT 2001 The Dallas Morning News, L.P.

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From Newcity Chicago

Most of us think of mathematics as some sort of inaccessible (and sometimes beautiful) language that's hidden within nature, only to be uncovered and plucked out through hours of rigorous and painful scholarship. Mathematics seems to be a language that transcends everyday reality, something outside and without us.
Apparently, nothing could be farther from the truth. In "Where Mathematics Comes From," George Lakoff and Raphael Nunez allay our misplaced xenophobic fear. In the tradition of cognitive science's investigation of the "embodiment of the human mind," Lakoff and Nunez take to task a wealth of common assumptions about mathematical "ideas" and, one by one, turn them all on their heads. For starters, the number line isn't actually a line. And the numbers don't actually fill it. It's not for the joy of bashing the king of sciences that Lakoff and Nunez dismantle common myths. The dissection allows us to see how math actually works, and how, as humans, we can possibly understand the ideas.
What's being espoused is mind-based mathematics, i.e., a math based on our innate ability to perform rudimentary arithmetic and subitization as infants, on basic cognitive tools like conceptual metaphors, blends, prototypes, frames, etc., and the concept of an embodied mind. Lakoff and Nunez try to show us that math doesn't just consist of moving obscure symbols around in a predetermined fashion, but that it encompasses concepts and ideas that can't be arrived at by standard ways of discussing math. The authors create a series of "idea analyses," explanations of how arithmetic operations are conceived of in terms of collecting objects. And how our concept of actual infinity is based on a series of finite processes-after all, how else could we conceive of the infinite when we, and all we encounter, are undeniably finite. Mathematics is actually a tool that we've constructed, not the language of the universe.
The joy in reading George Lakoff rubs off from his own joy in writing books about sophisticated and non-trivial topics. Keeping it readable for a wide audience -- no matter what the education level-he always takes great pain to achieve clarity. Lakoff's egalitarian approach is appropriate, considering the book takes to task the obfuscatory nature of mathematical symbol-mongering, but also because there's a secondary educational motive to the book: Lakoff and Nunez believe there's a better way to teach math. The proposed way teaches not only that theorems are true, but also why they're true. Which, of course, can only be done if we can generate a new understanding of this device we've created.
Reviewed by Nathan Matteson.

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From C&RL News

"Where Mathematics Comes From" is a good read for number buffs. Linguist Lakoff and psychologist Núñez contend that mathematics is rooted in everyday human cognitive activity instead of some transcendent Platonist netherworld. To prove their point, they take the reader on a metaphor-filled examination of basic arithmetic, algebra, infinity, and space-time that may put off arithmophobes and old fashioned prepostmodernists, but will excite anyone who thrills to such terminology as "discretized number-line blend," the "hierarchy of transfinite cardinals," and "fictive motion." A completely different way to look at the origin and structure of mathematical concepts.

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From American Library Association

With this ambitious book, Lakoff and Nunez hope to launch a whole new discipline: a cognitive science of mathematics. And they bid fair to bring it off, showing how all mathematical ideas--from simple counting to calculus--can be traced to the discrete workings of the human brain, and not to some transcendent realm of Platonic ideals. This approach to mathematics holds a number of surprises, as even ordinary arithmetic dissolves into conceptual metaphors grounded in the sensory-motor system. The entire panoply of mathematical symbols and calculations--precise and consistent--thus reflects the evolutionary history of brain neurons. Cognitive science can place even that most daunting of mathematical mysteries--infinity--within the observable human mind, explaining it as an aspect metaphor lodged deep in the unconscious. Similar reasoning can also account for the cultural plasticity of mathematics, which appears in one guise among the Mayans and a quite different one among the Chinese. A pioneering work of singular importance for mathematicians and psychologists alike--and of definite appeal to general readers with interest in those subjects. Bryce Christensen
Copyright © American Library Association. All rights reserved.

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From the Back Cover

"Where Mathematics Comes From is an unprecedented attempt to base advanced mathematics on the working of the brain. It may well be the beginning of a whole new direction in the philosophy of mathematics. It is beautifully written and a pleasure to read."
-Reuben Hersh, Professor of Mathematics, University of New Mexico; co-author of The Mathematical Experience, author of What is Mathematics Really?

"an unusually provocative naturalistic treatment of the foundations of mathematics based on lived experience and metaphor. It should force mathematicians to think harder about the foundations of their discipline."
-Felix Browder, President of the American Mathematical Society; University Professor of Mathematics at Rutgers University

"This fascinating book pioneers the application of modern cognitive science to mathematics, building detailed support to the often-denied truth that mathematics is founded on human thought. By systematically attacking the central romantic mythology of mathematics, this book is certain to provoke heated controversy, but it has great promise, when the dust finally settles, to leave us better able to teach, to communicate and to think mathematics."
-Bill Thurston, Professor of Mathematics, University of California, Davis and Fields Medal winner

"This is an important book, which challenges some deep seated beliefs about the nature of mathematics. If Lakoff and Nunez are right, and I believe they are, then what they say has significant implications for mathematics education."
-Keith Devlin, author of The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip.

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