Table Of ContentThe Art of
Mathematics
The Art 01
Ma thema tics
Jerry P. King
Springer Science+Business Media, LLC
Llbrary of Congrass Cataloglng-In-Publlcatlon Data
King, Jerry P.
The art of mathe.atles I Jerry P. King.
p. e •.
Ineludes blbllographleal references and Index.
I. Mathnatles. 1. Tltle.
QA3S.K47 1992
510--de20 91-4583S
CIP
Lines from "Easter 1916" are reprinted with permission of Macmillan Publishing Company from
The'Poems of W. B. Yeats: A New Edition, edited by Richard J. Finneran. Copyright 1924 by
Macmillan Publishing Company, renewed 1952 by Bertha Georgie Yeats.
Excerpts from "Mathematics and Creativity" by Alfred Adler reprinted by permission; Cl 1972
Tbe New Yorker Magazine, Inc.
Excerpts from "1\ Conservative Crack-Up?" by R. Emett Tyrrell, Jr. reprinted by permission of
The Wall Street Journal.
Excerpts from "Poet in Prose" by Doug Anderson copyright Cl 1991 by Tbe New York Times
Company. Reprinted by permission.
Excerpt from Fallingby James Dickey reprinted from Poems 1957~7Cl 1967 by James Dickey,
Wesleyan University Press. By permission of University Press ofNew England.
Excerpts from AStroll with William James by Jacques Barzun reprinted by permission of
HarperCollins Publishers.
Excerpts from The Two Cultures and a Second Look by C. P. Snow and A Mathematicians
Apology by G. H. Hardy reprinted by permission of Cambridge University Press.
Excerpts from "Tbe Mathematical Unconscious" by Seymore Papert reprinted from On
Aesthetics in Science, edited by Judith Wechsler Cl 1988 by Birkhäuser, with permission of
Springer-Verlag.
Excerpts from The Closing of the American Mind by Allan Bloom reprinted with permission of
Simon & Schuster.
Excerpts from Why the Professor Can't Teach by Morris Kline and Bridges to Infinity by
Michael Guillen reprinted with permission of St. Martin's Press.
First Printing - March 1992
Second Printing - March 1993
ISBN 978-0-306-44129-5 ISBN 978-1-4899-6339-0 (eBook)
DOI 10.1007/978-1-4899-6339-0
© 1992 Jerry P. King
Originally published by Plenum Press in 1992.
Softcover reprint of the hardcover 1s t edition 1992
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, e1ectronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
Contents
Introduction ........................................ 1
1. Tbe Unexpected ...................................... 13
The Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Accidents ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Trivialities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2. Pure Matbematics .................................... 25
Truth ................................................ 29
Research ............................................ 34
Creation ............................................. 41
3. Numbers............................................. 47
Axioms .............................................. 52
The Integers ......................................... 63
The Rational Numbers ............................... 64
The Real Numbers ................................... 67
The Complex Numbers............................... 73
Two Problems ....................................... 87
v
VI CONTENTS
4. Applied Mathematics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
Errors .............................................. 105
Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Applicable Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5. Aesthetics .......................................... 123
The Delicate Sieve .................................. 126
A Further Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
An Example ........................................ 133
Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
. Analytic Philosophy ................................ 148
The Artworld ....................................... 158
The Mathworld ..................................... 166
Truth and Beauty................................... 176
Principles .......................................... 179
Aesthetic Distance .................................. 194
Intuition Quiz Answers ............................. 208
6. Aristocracy ......................................... 21 1
Privilege ........................................... 221
Noblesse Oblige .................................... 226
Paradox ............................................ 233
7. The Two Cultures ................................... 247
Type M ............................................ 255
Uniqueness ......................................... 268
8. Great Things ...................................... 275
Low Art ............................................ 276
Value .............................................. 278
Promises ........................................... 286
9. Epilogue ............................................ 291
Notes .............................................. 297
Index............................................... 305
Introdudion
I~YI ears aga we summered in Vermont. We lived way
~ north in the kingdom of green mountains and taU
trees. The dirt road passing our house had remained un
changed since the revolutionary war and was so untraveled
that when a car came down we knew we had company. The
days drifted past quiet and easy and the loudest sound was
the occasional mooing of a cow in the pasture over the hill.
Quick fish moved in the c1ear, cold lake and the night sky
glistened with stars. Summer breezes tumed the grass.
One night, with the candles buming low, the wine
spent, and the moonlight just right, I said to my wife:
Vou are more beautiful than any wo man I have ever seen.
I was looking direct1y at her when I spoke. She looked
back. And she smiled softly. And she did not-I am happy
to say-think at that moment like a mathematician.
For had she done so, she would have conc1uded that
my compliment was non sense and nowhere near anything
2 INTRODUCTION
that could be called truth. For, if my statement were true,
then it would follow that she-being more beautiful than
any woman I had ever seen while simultaneously being the
object of my amorous gaze-would be more beautiful than
herself which, of course, is impossible. Had she examined
my remark with the cold light of precision she would
have found it meaningless. And she would have broken
the mood.
But she did not analyze. She knew what I meant.
And so did I. I opened another bottle of wine and blew
out the candles.
Mathematicians, unlike lovers, loathe imprecision.
Precision is the hallmark of mathematics and you can nei
ther talk mathematics nor write mathematics without first
committing yourself wholly to the strict adherence to rules
of inference and exposition. And when you do you will
capture the attention of the mathematicians. And for the
proper care and handling oftheir subject they will also give
you their appreciation. But that is all you are likely to get. If
you want more, candlelight and wine will serve you better
than precision.
I am writing-not mathematics-but about mathemat
ics. And the gap between these two notions stretches wide
as the distance between "physics" and "metaphysics." You
write about mathematics for nonmathematicians. Mathe
maticians, like Prufrock's mermaids, sing mathematics
only each to each. They do not sing for uso
This book is not written for mathematicians. Instead,
it is aimed toward an audience of those people for whom
art, music, and literature are common pursuits but for
whom mathematics may stand as a deep mystery. Or as
something horrid-a thing to be avoided like a nightmare.
Mathematicians know something the rest of us do not.
INTRODUCTION 3
They know that mathematics, rightly viewed, possesses an
aesthetic value as clearly defined as that of poetry. But no
one else knows this. This knowledge remains locked deep
inside the closed aristocracy of mathematicians.
Humanists find pleasure inside concert halls, art gal
leries, and great books but flee from mathematics like peas
ants racing from Frankenstein's monster. Meanwhile, the
aesthetic experience of mathematics both excites and
soothes mathematicians. The excitement brings them to
the subject and the soothing becomes addictive and draws
them back again and again. Their experience exactly dupli
cates that which time and again returns the humanist to the
Venus de Milo in the Louvre or to the impressionist pic
tures in the Musee d'Orsay.
Mathematicians also go to galleries and concerts-not
as often as they profess-but frequently. The aesthetic ex
periences ofthe humanists are open to the mathematicians.
But not the other way around. The aesthetic pleasures of
mathematics are unavailable to the humanists. Unavail
able not because the aesthetics lie beyond their comprehen
sion but because the right view of mathematics has been
kept hidden from them.
The book intends to expose the view, to bring the se
cret out into the light. And to show that one's intellectual
and aesthetic life cannot be complete unless it includes an
appreciation of the power and the beauty of mathematics.
Simply put, aesth-etic and intellectual fulfillment requires
that you know about mathematics.
The secrecy surrounding the aesthetics of mathematics
results partly from the fact that mathematicians do not like
to talk about mathematics. Neither do they like to write
about mathematics. Nor to read about mathematics.
Mathematicians like to do mathematics. And by doing
4 INTRODUCTION
mathematics they mean doing research in mathematics.
And by research they mean the creation of new mathemat
ics. They mean exactly this, and nothing more.
The reader must understand this distinction c1early: to
write about mathematics is to write exposition or descrip
tion with mathematics as the subject. To do mathematics is
to do research, which means to create mathematics as a
poet creates poems.
When mathematicians do mathematics they publish it
in journals. But what they write is mathematics and is not
about mathematics. They do not write exposition. And the
mathematics research they write is read, if at all, only by
other mathematicians. Consequently, there exists only a
sparse literature about mathematics and this is one reason
that otherwise knowledgeable people are completely un
aware ofthe existence ofhundreds ofmathematics research
journals and of the thousands of mathematicians who pub
lish in them.
Moreover, most ofthis literature is trivial because it is
written not by mathematicians but rather by outsiders who
fai! to understand either mathematics or its nature or why
people would want to do it. Occasionally, areal mathema
tician, for one reason or another, will produce a book about
mathematics. But, invariably, he will confess up front that
what he is doing is not his real work. Even though he knows
the expository book will reach more people than all his
research papers combined, the mathematician will assure
the reader that the book possesses little value when com
pared with, say, his most recent three page artic1e published
in the Journal ofthe American Mathematical Society.
A famous mathematician named G. H. Hardy once
wrote a book about mathematics. He was, however, made
so contrite by the process that at the outset he apologized to
the reader for having written. He, in fact, called the book A
