An imprint of Pearson Education, Inc.
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Foreword, Notes, Afterword and Further Readings © 2005 by Pearson
Education, Inc.© 1930, 1933, 1939, and 1954 by the Macmillan
Company
This edition is a republication of the 4th edition of Number, originally
published by Scribner, an Imprint of Simon & Schuster Inc.
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Printed in the United States of America
First Printing: March, 2005
Library of Congress Number: 2004113654
Pi Press books are listed at www.pipress.net.
ISBN 0-13-185627-8
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Contents
Foreword vii
Editor's Note xiv
Preface to the Fourth Edition xv
Preface to the First Edition xvii
1. Fingerprints 1
2. The Empty Column 19
3. Number-lore 37
4. The Last Number 59
5. Symbols 79
6. The Unutterable 103
7. This Flowing World 125
8. The Art of Becoming 145
9. Filling the Gaps 171
10. The Domain of Number 187
11. The Anatomy of the Infinite 215
12. The Two Realities 239
Appendix A. On the Recording of Numbers 261
Appendix B. Topics in Integers 277
Appendix C. On Roots and Radicals 303
Appendix D. On Principles and Arguments 327
Afterword 343
Notes 351
Further Readings 373
Index 385
Foreword
The book you hold in your hands is a many-stranded meditation
on Number, and is an ode to the beauties of mathematics.
This classic is about the evolution of the Number concept. Yes:
Number has had, and will continue to have, an evolution. How did
Number begin? We can only speculate.
Did Number make its initial entry into language as an adjective?
Three cows, three days, three miles. Imagine the exhilaration
you would feel if you were the first human to be struck with the
startling thought that a unifying thread binds “three cows” to “three
days,” and that it may be worthwhile to deal with their common
three-ness. This, if it ever occurred to a single person at a single
time, would have been a monumental leap forward, for the disembodied
concept of three-ness, the noun three, embraces far more
than cows or days. It would also have set the stage for the comparison
to be made between, say, one day and three days, thinking of
the latter duration as triple the former, ushering in yet another
view of three, in its role in the activity of tripling; three embodied,
if you wish, in the verb to triple.
Or perhaps Number emerged from some other route: a form
of incantation, for example, as in the children’s rhyme “One, two,
buckle my shoe….”
However it began, this story is still going on, and Number,
humble Number, is showing itself ever more central to our understanding
of what is. The early Pythagoreans must be dancing in
If I were someone who had a yen to learn about math, but
their caves.
never had the time to do so, and if I found myself marooned on
that proverbial “desert island,” the one book I would hope to have
along is, to be honest, a good swimming manual. But the second
book might very well be this one. For Dantzig accomplishes these
essential tasks of scientific exposition: to assume his readers have
no more than a general educated background; to give a clear and
vivid account of material most essential to the story being told; to
tell an important story; and—the task most rarely achieved of all—
to explain ideas and not merely allude to them.
One of the beautiful strands in the story of Number is the
manner in which the concept changed as mathematicians expanded
the republic of numbers: from the counting numbers
1, 2, 3,…
to the realm that includes negative numbers, and zero
… –3, –2, –1, 0, +1, +2, +3,…
and then to fractions, real numbers, complex numbers, and, via a
different mode of colonization, to infinity and the hierarchy of
infinities. Dantzig brings out the motivation for each of these augmentations:
There is indeed a unity that ties these separate steps
into a single narrative. In the midst of his discussion of the expansion
of the number concept, Dantzig quotes Louis XIV.When asked
what the guiding principle was of his international policy, Louis
XIV answered, “Annexation! One can always find a clever lawyer to
vindicate the act.”But Dantzig himself does not relegate anything to
legal counsel. He offers intimate glimpses of mathematical birth
pangs, while constantly focusing on the vital question that hovers
over this story: What does it mean for a mathematical object to
exist? Dantzig, in his comment about the emergence of complex
numbers muses that “For centuries [the concept of complex numbers]
figured as a sort of mystic bond between reason and imagination.”
He quotes Leibniz to convey this turmoil of the intellect:
“[T]he Divine Spirit found a sublime outlet in that wonder of
analysis, that portent of the ideal world, that amphibian between
being and not-being, which we call the imaginary root of negative
unity.” (212)
Dantzig also tells us of his own early moments of perplexity:
“I recall my own emotions: I had just been initiated into the mysteries
of the complex number. I remember my bewilderment: here
were magnitudes patently impossible and yet susceptible of
manipulations which lead to concrete results. It was a feeling of
dissatisfaction, of restlessness, a desire to fill these illusory creatures,
these empty symbols, with substance. Then I was taught to
interpret these beings in a concrete geometrical way. There came
then an immediate feeling of relief, as though I had solved an
enigma, as though a ghost which had been causing me apprehension
turned out to be no ghost at all, but a familiar part of my
environment.” (254)
The interplay between algebra and geometry is one of the
grand themes of mathematics. The magic of high school analytic
geometry that allows you to describe geometrically intriguing
curves by simple algebraic formulas and tease out hidden properties
of geometry by solving simple equations has flowered—in
modern mathematics—into a powerful intermingling of algebraic
and geometric intuitions, each fortifying the other. René Descartes
proclaimed: “I would borrow the best of geometry and of algebra
and correct all the faults of the one by the other.” The contemporary
mathematician Sir Michael Atiyah, in comparing the glories of
geometric intuition with the extraordinary efficacy of algebraic
“Algebra is the offer made by the devil to the mathematician. The
methods, wrote recently:
devil says: I will give you this powerful machine, it will answer any
question you like. All you need to do is give me your soul: give up
geometry and you will have this marvelous machine. (Atiyah, Sir
Michael. Special Article: Mathematics in the 20th Century. Page 7.
Bulletin of the London Mathematical Society, 34 (2002) 1–15.)”
It takes Dantzig’s delicacy to tell of the millennia-long
courtship between arithmetic and geometry without smoothing
out the Faustian edges of this love story.
In Euclid’s Elements of Geometry, we encounter Euclid’s definition
of a line: “Definition 2. A line is breadthless length.”
Nowadays, we have other perspectives on that staple of plane
geometry, the straight line. We have the number line, represented
as a horizontal straight line extended infinitely in both directions
on which all numbers—positive, negative, whole, fractional, or
irrational—have their position. Also, to picture time variation, we
call upon that crude model, the timeline, again represented as a
horizontal straight line extended infinitely in both directions, to
stand for the profound, ever-baffling, ever-moving frame of
past/present/futures that we think we live in. The story of how
these different conceptions of straight line negotiate with each
other is yet another strand of Dantzig’s tale.
Dantzig truly comes into his own in his discussion of the relationship
between time and mathematics. He contrasts Cantor’s
theory, where infinite processes abound, a theory that he maintains
is “frankly dynamic,” with the theory of Dedekind, which he refers
to as “static.” Nowhere in Dedekind’s definition of real number,
says Dantzig, does Dedekind even “use the word infinite explicitly,
or such words as tend, grow, beyond measure, converge, limit, less
At this point, reading Dantzig’s account, we seem to have come
than any assignable quantity, or other substitutes.”
to a resting place, for Dantzig writes:
“So it seems at first glance that here [in Dedekind’s formulation of
real numbers] we have finally achieved a complete emancipation
of the number concept from the yoke of time.” (182)
To be sure, this “complete emancipation” hardly holds up to
Dantzig’s second glance, and the eternal issues regarding time and
its mathematical representation, regarding the continuum and its
relationship to physical time, or to our lived time—problems we
have been made aware of since Zeno—remain constant companions
to the account of the evolution of number you will read in this
book.
Dantzig asks: To what extent does the world, the scientific
world, enter crucially as an influence on the mathematical world,
and vice versa?
“The man of science will acts as if this world were an absolute
whole controlled by laws independent of his own thoughts or act;
but whenever he discovers a law of striking simplicity or one of
sweeping universality or one which points to a perfect harmony in
the cosmos, he will be wise to wonder what role his mind has
played in the discovery, and whether the beautiful image he sees in
the pool of eternity reveals the nature of this eternity, or is but a
reflection of his own mind.” (242)
Dantzig writes:
“The mathematician may be compared to a designer of garments,
who is utterly oblivious of the creatures whom his garments may
fit. To be sure, his art originated in the necessity for clothing such
creatures, but this was long ago; to this day a shape will occasionally
appear which will fit into the garment as if the garment had
been made for it. Then there is no end of surprise and of delight!”
This bears some resemblance in tone to the famous essay of the
(240)
physicist Eugene Wigner, “The Unreasonable Effectiveness of
Mathematics in the Natural Sciences,” but Dantzig goes on, by
offering us his highly personal notions of subjective reality and
objective reality. Objective reality, according to Dantzig, is an
impressively large receptacle including all the data that humanity
has acquired (e.g., through the application of scientific instruments).
He adopts Poincaré’s definition of objective reality, “what
is common to many thinking beings and could be common to all,”
to set the stage for his analysis of the relationship between Number
and objective truth.
Now, in at least one of Immanuel Kant’s reconfigurations of
those two mighty words subject and object, a dominating role is
played by Kant’s delicate concept of the sensus communis. This sensus
communis is an inner “general voice,” somehow constructed
within each of us, that gives us our expectations of how the rest of
humanity will judge things.
The objective reality of Poincaré and Dantzig seems to require,
similarly, a kind of inner voice, a faculty residing in us, telling us
something about the rest of humanity: The Poincaré-Dantzig
objective reality is a fundamentally subjective consensus of what is
commonly held, or what could be held, to be objective. This view
already alerts us to an underlying circularity lurking behind many
discussions regarding objectivity and number, and, in particular
behind the sentiments of the essay of Wigner. Dantzig treads
around this lightly.
My brother Joe and I gave our father, Abe, a copy of Number:
The Language of Science as a gift when he was in his early 70s. Abe
had no mathematical education beyond high school, but retained
an ardent love for the algebra he learned there. Once, when we were
quite young, Abe imparted some of the marvels of algebra to us:
“I’ll tell you a secret,” he began, in a conspiratorial voice. He proceeded
to tell us how, by making use of the magic power of the
add one to it you get 11. I was quite a literal-minded kid and really
cipher X, we could find that number which when you double it and
thought of X as our family’s secret, until I was disabused of this
attribution in some math class a few years later.
Our gift of Dantzig’s book to Abe was an astounding hit. He
worked through it, blackening the margins with notes, computations,
exegeses; he read it over and over again. He engaged with
numbers in the spirit of this book; he tested his own variants of the
Goldbach Conjecture and called them his Goldbach Variations. He
was, in a word, enraptured.
But none of this is surprising, for Dantzig’s book captures both
soul and intellect; it is one of the few great popular expository classics
of mathematics truly accessible to everyone.
—Barry Mazur
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