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Contents
Acknowledgments ix
��� ri
IntrodWuhcyCt oigonni:St ciiveMena ctett eoMr ast hematic1s
Partl
THEE MBODIMEONFBT A SIACR ITHMETIC
1 ThBer aIinnn'Asar tiet hmetic 15
2 BAr iIenft rodtuotc htCeio ognn itivoef SEtcmhibeeo ndMciieen dd2 7
3 EmbodAireidt hmTehGteri ocu:n Mdeitnagp hors 50
4 WheDroet hLea wosAf r ithCmoemtei cF rom? 77
ParItI
ALGEBRLAO,G IACN,D S ETS
5 EsseanncAdel gebra 107
6 BoolMee'tsa pChloarsa:sn eSdsy mboLloigci c 121
7 SeatnsHd y persets 140
ParItl l
THEE MBODIMEONFIT N FINITY
8 ThBea sMiect apohfIo nrfi nity 155
9 Real NuamnLbdie mrist s 181
VII
VIII CONTENTS
10 Transfinite Numbers 208
11 Infinitesimals 223
Part IV
BANNING SPACE AND MOTION: THE DISCRETIZATION
PROGRAM THAT SHAPED MODERN MATHEMATICS
12 Points and the Continuum 259
13 Continuity for Numbers: The Triumph of Dedekind's Metaphors 292
14 Calculus Without Space or Motion: Weierstrass's
Metaphorical Masterpiece 306
Le tnroorum and:
A CLASSIC PARADOX OF INFINITY 325
PartV
IMPLICATIONS FOR THE PHILOSOPHY OF MATHEMATICS
15 The Theory of Embodied Mathematics 337,
16 The Philosophy of Embodied Mathematics 364
Part VI
e'ITi + 1 = 0
A CASE STUDY OF THE COGNITIVE STRUCTURE
OF CLASSICAL MATHEMATICS
Case Study 1. Analytic Geometry and Trigonometry 383
Case Study 2. What Is e? 399
Case Study 3. What Is i? 420
Case Study 4. e'ITi + 1 = 0-How the Fundamental Ideas of
Classical Mathematics Fit Together 433
References 453
Index 473
Acknowled ments
g
T
without a lot of help, espe
HIS BOOK WOULD NOT HAVE BEEN POSSIBLE
cially from mathematicians and mathematics students, mostly at the
University of California at Berkeley.
Our most immediate debt is to Reuben Hersh, who has long been one of our
heroes and who has supported this endeavor since its inception.
We have been helped enormously by conversations with mathematicians, es
pecially Ferdinando Arzarello, Elwyn Berlekamp, George Bergman, Felix Brow
der, Martin Davis, Joseph Goguen, Herbert Jaeger, William Lawvere, Leslie
Lamport, Lisa Lippincott, Giuseppe Longo, Robert Osserman, Jean Petitot,
David Steinsalz, William Thurston, and Alan Weinstein. Other scholars con
cerned with mathematical cognition and education have made crucial contri
butions to our understanding, especially Mariolina Bartolini Bussi, Paolo Boero,
Janete Bolite Frant, Stanislas Dehaene, Jean-Louis Dessalles, Laurie Edwards,
Lyn English, Gilles Fauconnier, Jerry Feldman, Deborah Forster, Cathy Kessel,
Jean Lassegue, Paolo Mancosu, Maria Alessandra Mariotti, Joao Filipe Matos,
Srini Narayanan, Bernard Plancherel, Jean Retschitzki, Adrian Robert, and
Mark Turner. In spring 1997 we gave a graduate seminar at Berkeley, called
"The Metaphorical Structure of Mathematics: Toward a Cognitive Science of
Mathematical Ideas," as a way of approaching the task of writing this book. We
are grateful to the students and faculty in that seminar, especially Mariya Brod
sky, Mireille Broucke, Karen Edwards, Ben Hansen, Ilana Horn, Michael Kleber,
Manya Raman, and Lisa Webber. We would also like to thank the students in
George Lakoff's course on The Mind and Mathematics in spring 2000, expe
cially Lydia Chen, Ralph Crowder, Anton Dochterman, Markku Hannula, Jef
frey Heer, Colin Holbrook, Amit Khetan, Richard Mapplebeck-Palmer,
Matthew Rodriguez, Eric Scheel, Aaron Siegel, Rune Skoe, Chris Wilson, Grace
Wang, and Jonathan Wong. Brian Denny, Scott Nailor, Shweta Narayan, and
IX
X ACKNOWLEDGMENTS
Katherine Talbert helped greatly with their detailed comments on a preliminary
version of the manuscript.
A number of readers have pointed out errors in the first printing, some typo
graphical, some affecting content. We extend our thanks for this assistance to
Joseph Auslander, Bonnie Gold, Alan Jennings, Hector Lomeli, Michael Reeken,
and Norton Starr.
We are not under the illusion that we can thank our partners, Kathleen
Frumkin and Elizabeth Beringer, sufficiently for their immense support in this
enterprise, but we intend to work at it.
The great linguist and lover of mathematics James D. McCawley, of the Uni
versity of Chicago, died unexpectedly before we could get him the promised
draft of this book. As someone who gloried in seeing dogma overturned and who
came to intellectual maturity having been taught that mathematics provided
foundations for linguistics, he would have delighted in the irony of seeing argu
ments for the reverse.
Just before the publication of this paperback edition, our friend the brilliant
neuroscientist Francisco J. Varela, of the Laboratory of Cognitive Neurosciences
and Brain Imaging in Paris, died after a long illness. His support had been fun
damental to our project, and we had looked forward to many more years of fruit
ful dialog, which his untimely death has brought to an abrupt end. His deep and
pioneering understanding of the embodied mind will continue to inspire us, as
it has for the past two decades.
It has been our great good fortune to be able to do this research in the mag
nificent, open intellectual community of the University of California at Berke
ley. We would especially like to thank the Institute of Cognitive Studies and the
International Computer Science Institute for their support and hospitality over
many years.
Rafael N\'.1ftez would like to thank the Swiss National Science Foundation for
fellowship support that made the initial years of this research possible. George
Lakoff offers thanks to the university's Committee on Research for grants that
helped in the preparation and editing of the manuscript.
Finally, we can think of few better places in the world to brainstorm and find
sustenance than O Chame, Cafe Fanny, Bistro Odyssia, Cafe N efeli, The Musi
cal Offering, Cafe Strada, and Le Bateau Ivre.
Preface
W
E ARE COGNITIVE SCIENTISTS-a linguist and a psychologist-each
with a long-standing passion for the beautiful ideas of mathematics.
As specialists within a field that studies the nature and structure of ideas,
we realized that despite the remarkable advances in cognitive science and a long
tradition in philosophy and history, there was still no discipline of mathe
matical idea analysis from a cognitive perspective-no cognitive science of
mathematics.
With this book, we hope to launch such a discipline.
A discipline of this sort is needed for a simple reason: Mathematics is deep,
fundamental, and essential to the human experience. As such, it is crying out
to be understood.
It has not been.
Mathematics is seen as the epitome of precision, manifested in the use of
symbols in calculation and in formal proofs. Symbols are, of course, just sym
bols, not ideas. The intellectual content of mathematics lies in its ideas, not in
the symbols themselves. In short, the intellectual content of mathematics does
not lie where the mathematical rigor can be most easily seen-namely, in the
symbols. Rather, it lies in human ideas.
But mathematics by itself does not and cannot empirically study human
ideas; human cognition is simply not its subject matter. It is up to cognitive sci
ence and the neurosciences to do what mathematics itself cannot do-namely,
apply the science of mind to human mathematical ideas. That is the purpose of
this book.
One might think that the nature of mathematical ideas is a simple and obvi
ous matter, that such ideas are just what mathematicians have consciously
taken them to be. From that perspective, the commonplace formal symbols do
as good a job as any at characterizing the nature and structure of those ideas. If
that were true, nothing more would need to be said.
XI
XII PREFACE
But those of us who study the nature of concepts within cognitive science
know, from research in that field, that the study of human ideas is not so sim
ple. Human ideas are, to a large extent, grounded in sensory-motor experience.
Abstract human ideas make use of precisely formulatable cognitive mecha
nisms such as conceptual metaphors that import modes of reasoning from sen
sory-motor experience. It is always an empirical question just what human
ideas are like, mathematical or not.
The central question we ask is this: How can cognitive science bring sys
tematic scientific rigor to the realm of human mathematical ideas, which lies
outside the rigor of mathematics itself? Our job is to help make precise what
mathematics itself cannot-the nature of mathematical ideas.
Rafael Nunez brings to this effort a background in mathematics education,
the development of mathematical ideas in children, the study of mathematics
in indigenous cultures around the world, and the investigation of the founda
tions of embodied cognition. George Lakoff is a major researcher in human con
ceptual systems, known for his research in natural-language semantics, his
work on the embodiment of mind, and his discovery of the basic mechanisms
of everyday metaphorical thought.
The general enterprise began in the early 1990s with the detailed analysis by
one of Lakoff's students, Ming Ming Chiu (now a professor at the Chinese Uni
versity in Hong Kong), of the basic system of metaphors used by children to
comprehend and reason about arithmetic. In Switzerland, at about the same
time, Nunez had begun an intellectual quest to answer these questions: How
can human beings understand the idea of actual infinity-infinity conceptual
ized as a thing, not merely as an unending process? What is the concept of ac
tual infinity in its mathematical manifestations-points at infinity, infinite
sets, infinite decimals, infinite intersections, transfinite numbers, infinitesi
mals? He reasoned that since we do not encounter actual infinity directly in the
world, since our conceptual systems are finite, and since we have no cognitive
mechanisms to perceive infinity, there is a good possibility that metaphorical
thought may be necessary for human beings to conceptualize infinity. If so, new
results about the structure of metaphorical concepts might make it possible to
precisely characterize the metaphors used in mathematical concepts of infinity.
With a grant from the Swiss NSF, he came to Berkeley in 1993 to take up this
idea with Lakoff.
We soon realized that such a question could not be answered in isolation. We
would need to develop enough of the foundations of mathematical idea analysis
so that the question could be asked and answered in a precise way. We would need
to understand the cognitive structure not only of basic arithmetic but also of sym-
