Number

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Pi Press

New York

NUMBER

The Language of Science

Tobias

Dantzig

Edited by

Joseph Mazur

Foreword by

Barry Mazur

The Masterpiece Science Edition

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An imprint of Pearson Education, Inc.

1185 Avenue of the Americas, New York, New York 10036

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.

Pi Press offers discounts for bulk purchases. For more information,

please contact U.S. Corporate and Government Sales, 1-800-382-3419,

[email protected]. For sales outside the U.S., please

contact International Sales at [email protected].

Company and product names mentioned herein are the trademarks or

registered trademarks of their respective owners.

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

Pearson Education LTD.

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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

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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

Contents vi

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Foreword

The book you hold in your hands is a many-stranded medi￾tation on Number, and is an ode to the beauties of mathe￾matics.

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 adjec￾tive? 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 disem￾bodied concept of three-ness, the noun three, embraces far more

than cows or days. It would also have set the stage for the compar￾ison 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 under￾standing of what is. The early Pythagoreans must be dancing in

their caves.

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viii NUMBER

If I were someone who had a yen to learn about math, but

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 expand￾ed 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 aug￾mentations: There is indeed a unity that ties these separate steps

into a single narrative. In the midst of his discussion of the expan￾sion 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 num￾bers] figured as a sort of mystic bond between reason and imagina￾tion.” He quotes Leibniz to convey this turmoil of the intellect:

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Foreword ix

“[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 mys￾teries 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 crea￾tures, 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 apprehen￾sion 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 proper￾ties 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 contempo￾rary mathematician Sir Michael Atiyah, in comparing the glories of

geometric intuition with the extraordinary efficacy of algebraic

methods, wrote recently:

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“Algebra is the offer made by the devil to the mathematician. The

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 defin￾ition 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 rela￾tionship 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

than any assignable quantity, or other substitutes.”

x NUMBER

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Foreword xi

At this point, reading Dantzig’s account, we seem to have come

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 compan￾ions 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 occasion￾ally 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!”

(240)

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This bears some resemblance in tone to the famous essay of the

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 instru￾ments). 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 sen￾sus 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 pro￾ceeded to tell us how, by making use of the magic power of the

cipher X, we could find that number which when you double it and

xii NUMBER

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Foreword xiii

add one to it you get 11. I was quite a literal-minded kid and really

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, computa￾tions, 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 clas￾sics of mathematics truly accessible to everyone.

—Barry Mazur

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Editor’s Note to the Masterpiece

Science Edition

The text of this edition of Number is based on the fourth edi￾tion, which was published in 1954. A new foreword, after￾word, endnotes section, and annotated bibliography are

included in this edition, and the original illustrations have been

redrawn.

The fourth edition was divided into two parts. Part 1,

“Evolution of the Number Concept,” comprised the 12 chapters

that make up the text of this edition. Part 2, “Problems Old and

New,” was more technical and dealt with specific concepts in depth.

Both parts have been retained in this edition, only Part 2 is now set

off from the text as appendixes, and the “part” label has been

dropped from both sections.

In Part 2, Dantzig’s writing became less descriptive and more

symbolic, dealing less with ideas and more with methods, permit￾ting him to present technical detail in a more concise form. Here,

there seemed to be no need for endnotes or further commentaries.

One might expect that a half-century of advancement in mathe￾matics would force some changes to a section called “Problems Old

and New,” but the title is misleading; the problems of this section

are not old or new, but are a collection of classic ideas chosen by

Dantzig to show how mathematics is done.

In the previous editions of Number, sections were numbered

within chapters. Because this numbering scheme served no func￾tion other than to indicate a break in thought from the previous

paragraphs, the section numbers were deleted and replaced by a

single line space.

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