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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 (coauthor, with Mark Johnson, of Metaphors
We Live By) and psychologist Núñez
(coeditor of Reclaiming Cognition) brings two decades of insights
from cognitive science to bear on the nature of human mathematical thought,
beginning with the basic, preverbal 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 pressbut 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: imageschemas, 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
etothepowerix, allowing us to characterize in cognitive terms what
Euler's equation etothepower[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
abstractionLakoff 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 conceptsa 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 thoughtprovoking. 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 realword
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 mindbased 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 processesafter
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 nontrivial topics. Keeping it readable for a
wide audience  no matter what the education levelhe always takes great
pain to achieve clarity. Lakoff's egalitarian approach is appropriate,
considering the book takes to task the obfuscatory nature of mathematical
symbolmongering, 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
metaphorfilled examination of basic arithmetic, algebra, infinity, and
spacetime that may put off arithmophobes and old fashioned prepostmodernists,
but will excite anyone who thrills to such terminology as "discretized
numberline 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 ideasfrom simple counting to calculuscan
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 sensorymotor system. The entire panoply of
mathematical symbols and calculationsprecise and consistentthus reflects
the evolutionary history of brain neurons. Cognitive science can place
even that most daunting of mathematical mysteriesinfinitywithin 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 alikeand 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;
coauthor 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 oftendenied 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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