The Infinite in the Finite

539 Pages·1996·11.47 MB·English

by  Alistair Macintosh Wilson

THE INFINITE IN
THE FINITE
ALISTAIR MACINTOSH WILSON
Oxford New York Tokyo
OXFORD UNIVERSITY PRESS
1995

THE INFINITE IN THE FINITE
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Libra1y of Congress Cataloging in Publication Data
Wilson, Alistair Macintosh.
The infinite in the finite/Alistair Macintosh Wilson.
Includes bibliographical references and index.
1. Mathematics-Hist01y. I. Title.
QA21.W3~5 1995 510'.9-dc20 94-34260
ISBN 0-19-853950-9
For
Lynette Eileen
and
The Opal Talleys

PREFACE
I have never read prefaces to books. As a student I came to believe that
prefaces are the places where authors, relieved finally of the burden of their
books, parade their stables of pet hobby-horses. The purpose of this preface
is therefore not to bore the reader with my views on the teaching of
elementary mathematics or its history. It is simply to tell a few stories about
how I came to write this book, and to thank the people without whose help
it would never have come into existence.
In the mid-1950s, I was very interested, like most children my age, in
atomic bombs. Having read some of the popular accounts of the day, and
seen some of the propaganda films of various 'ban-the-bomb' organiza
tions, I was dissatisfied. I wanted to know how a nuclear weapon 'really'
worked. I asked my cousin Michael, at the time manfully struggling
through the Cambridge Mathematics Tripos. Michael told me that to
understand how an atom bomb worked I would have to learn a lot of
mathematics. After attempting unsuccessfully to teach me calculus, he
suggested I started with trigonometry. He gave me E. T. Bell's Men of
mathematics to read. I loved the stories of the great mathematicians, but
found I couldn't learn any mathematics from this book.
When I returned to school (Bedales) I went to the library to find a book
on trigonometry. Being very small for my age, I remember how huge
the library doors seemed. I emerged from the Bedales library with no
knowledge of trigonometry, but clutching one fact which I have never
forgotten; a radian is 5J017'44.8"'. I had no idea why mathematicians
should choose to measure angles in terms of this strange unit. Further trips
to the library unearthed another book which I enjoyed; W. W. Sawyer's
Mathematicians delight.
By then my ideas on how I wanted to learn mathematics had begun to
take concrete form. I wanted to learn mathematics as a story. Since no book
that I could find presented the subject in this way, I rejected them all with
the ferocity of childhood as 'a pack of trash'. I very quickly convinced
myself that I was no good at mathematics. At the same time, however,
I believed that behind the courses I sat through with rising irritation and
incomprehension, there was a 'real' mathematics out of whose living force
came the problems and solutions which seemed to me to appear at random.
Vlll Preface
I thought that this 'real' mathematics probably existed in the papers of the
great mathematicians whose names I'd read in Bell and Sawyer. These
papers were of course totally closed to me.
After a few years of searching for a book which would teach me
mathematics the way I wanted to learn it, I gave up. Obviously I had no
aptitude for the subject and that was that. The Ordinary and Advanced
level examinations of the Central Welsh Board only confirmed this fact, and
any lingering doubts on the matter were laid to rest by the 'ancillary
mathematics for physicists' courses at London University. My feeble
attempts to teach myself some 'modern' mathematics were effectively
blocked by the dragon of 'mathematese'.
But every once in a while over the years would come a projection from
the world of 'real' mathematics, reminding me that it was still there. One of
these occurred whilst I was finishing off my thesis at the Cambridge
Observatories. There was a small library in the old Maths Lab in which
I slept fairly regularly. One night, unable to sleep, I picked up Cornelius
Lanczos' Applied analysis. I was surprised to find I could read it with
pleasure. I remember telling my friend Gordon Worral 'I've found a
maths book we can read, Gordon!'. By then we had more or less accepted
that asking mathematicians for help was a waste of time. Trying to find
how to solve a Fredholm integral equation of the first kind numerically
we were told variously that 'It has a solution', 'The problem is mal pose'
(true), 'It's a contraction mapping'. None of which helped us determine
the run of temperature into the Sun's photosphere. My attempts to extract
from E. H. Linfoot some scheme via which I could teach myself
mathematics was an equal failure. I settled for reminiscences of Oxford
and Gottingen.
I began to think about writing a book on 2 August 1972. I was driving
up the road to Kitt Peak National Observatory in Arizona taking a very
interesting young lady I'd met in Tucson a month earlier to see the 'biggest
sun in the world'; the image of the McMath-Hulbert solar telescope. In the
course of our conversation the girl said 'I hate math'. I remember thinking
'You've never seen any real mathematics, no more than I have.'
By the autumn of 1972 I had convinced myself that I knew nothing and
that I ought to try to re-educate myself by reading original papers. On
9 October 1972 I went to the library at Goddard Space Flight Center to
begin this process. I tried to read Schrodinger's first paper on Wave
Mechanics. I found I could read the German but not the paper! Over the
next two and a half years I spent many evenings in Goddard library. Of the
8000 employees at Goddard only one used this facility out of office hours
during this period: Dr Chung Chieh Cheng.
In September 1975 I promised thv girl I had taken up the mountain that
I would write a book for her. I also foolhardily promised another American
Preface lX
friend Bob McDonald the same. These books went through more changes
of form than the Proteus. First there was a book on stellar atmosphere
theory. This expanded into a history of modern astrophysics which
contracted to a short history of physics, expanding again to a history of the
application of mathematics to physics. I remember hot-footing down the
Huntingdon Road in Cambridge in the summer of 1980 with my friend
Larry Falvello, to send a particularly bulky one of these manuscripts to a
publisher. None of them was ever published. By the mid-1980s it didn't
look as if I'd be able to keep my promise to the two American friends for
whom I'd promised to write books.
In early 1986 one of my Australian students Scott Simms turned up one
day and said 'Have you heard of this desk-top publishing?' I hadn't. He told
me that if I wrote a manuscript I could get it typed and bound and that
would be a book. This seemed an obvious way out of my dilemma. After
all I'd never said a 'published' book. So I wrote up a manuscript which was
typed by Scott and his sister. Another friend Mike Handley made a beautiful
job of the mathematical diagrams. We got the book bound and sent it off
to the lady in question. Six months or so later Oxford informed us they
were interested in publishing this book.
Various parts of this book were written for particular people. 'The
pyramid builders' and 'Theban mysteries' describing Africa's contribution
to the birth of mathematics are for Frederick Guidry. All references to
things Chinese are for Dr Chung Chieh Cheng. 'The philosopher's criticism'
is for Tony Vasaio. 'The thoughts of Zeus' is for Larry Falvello. 'The island
interlude' is for my wife Stormy, the girl I took up the mountain. The old
man wherever he appears in the book is my grandfather Dr J. M. Wilson,
as I remember him. The few mentions of topology are for my friend Marcus
Pinto. The section on Islam in Spain is for Larry's wife Milagros.
The chapters on Archimedes and Apollonius are attempts to learn
directly from the works of the masters. If they lead the reader to turn to
these works for themselves, then they are successful.
In addition to those mentioned above I would like to thank Ed
Kibblewhite, Aliposo Waquailiti, and Charles Wolff for their encourage
ment over the years. In Perth I must thank my friends at the Curtin Dome:
Shelagh Pascoe, George Larcher, Leeanne Sharples, and Leighton Hogan,
and also Barry and Joan Williams, Bruce and Haziyah Bell, Azelan and
Denise Groom, and Reg and Lily Pinto. As described above this book came
into existence due to the selfless work of Scott Simms and Mike Handley.
I thank them again here.
Finally, and most important of all, I would like to thank Mam, Ann, and
Hugh in Aberdare for encouraging me in the apparently at times forlorn
belief that something I wrote would eventually get published. I hope they
will enjoy this book.