Sách Breakthroughs in mathematics - PETER WOLFF

Breakthroughs in mathematics - PETER WOLFF

CONTENTS
INTRODUCTION
I GEOMETRY
CHAPTER ONE
Euclid-The Beginnings of Geometry
CHAPTER TWO
Lobachevski-Non-Euclidean
Geometry
CHAPTER THREE
Descartes-Geometry and Algebra Joined
II ARITHMETIC
CHAPTER FOUR
Archimedes-Numbers and Counting
CHAPTER FIVE
Dedekind-Irrational Numbers
CHAPTER SIX
Russell-The Definition of Number
II ADVANCED TOPICS
CHAPTER SEVEN
Euler-A New Branch of Mathematics : Topology
CHAPTER EIGHT
Laplace-The Theory of Probability
CHAPTER NINE
Boole-Algebra and Logic Joined
SUGGESTIONS FOR FURTHER REAIMNG
INDEX

INTRODUCTION
The nine mathematicians whose works are represented in
the following pages are among the most famous in the whole
history of mathematics. Each of them made a significant contribution
to the science-a contribution which changed the
succeeding course of the development of mathematics. That
is why we have called this book Breakthroughs in Mathematics.
Just as surely as there are technological breakthroughs
which change our way of living, so are there breakthroughs
in the pure sciences which have such an impact that they affeet
all succeeding thought.
The mathematicians whose works we examine bridge a span
of more than 2200 years, from Euclid, who lived and worked
in Alexandria around 300 B.C., to Bertrand Russell, whose
major mathematical work was accomplished in the fist years
of the twentieth century. These nine chapters survey the major
parts of mathematics; a great many of its branches are touched
on. We shall have occasion to deal with geometry, both Euclidean
and non-Euclidean, with arithmetic, algebra, analytic
geometry, the theory of irrationals, set-theory, calculus of
probability, and mathematical logic. Also, though this is a matter
of accident, the authors whose works we study come from
almost every important country in the West: from ancient
Greece and Hellenistic Rome, Egypt, France, Germany, Great
Britain, and Russia.
No collection of nine names could possibly include all the
great mathematicians. Let us just name some of the most famous
ones whom we had to ignore here: Apollonius of Perga,
Pierre de Fermat, Blaise Pascal, Sir Isaac Newton, Gottfried
Wilhelm Leibniz, Karl Friedrich Gauss, Georg Cantor, and
many, many others. There are a number of reasons why we
chose the particular authors and books represented. In part,
a choice such as this is, of course, based on subjective and
personal preferences. On objective grounds, however, we were
mainly interested in presenting treatises or parts of treatises
that would exemplify the major branches of mathematics, that
would be complete and understandable in themselves, and
that would not require a great deal of prior mathematical
knowledge. There is one major omission which we regret:
none of the works here deals with the calculus. The reason is
that neither Newton nor Leibniz (who simultaneously developed
modern calculus) has left us a short and simple treatise
on the subject. Newton, to be sure, devotes the beginning of
his Principia to the calculus, but unfortunately his treatment
of the matter is not easy to understand.
What is the purpose in presenting these excerpts and the
commentaries on them? Very simply, we want to afford the
reader who is interested in mathematics and in the history of
its development an opportunity to see great mathematical
minds at work. Most readers of this book will probably already
have read some mathematical books-in school if nowhere
else. But here we give the reader a view of mathematics as it
is being developed; he can follow the thought of the greatest
mathematicians as they themselves set it down. Most great
mathematicians are also great teachers of mathematics; certainly
these nine writers make every possible effort to make
their discoveries understandable to the lay reader. (The one
exception may be Descartes, who practices occasional deliberate
obscurity in order to show off his own brilliance.) Each
of these selections can be read independently of the others, as
an example of mathematical genius at work. Each selection
will make the reader acquainted with an important advance
in mathematics; and he will learn about it from the one person
best qualified to teach him-its discoverer.
Breakthroughs in Mathematics is not a textbook. It does
not aim at the kind of completeness that a textbook possesses.
Rather it aims to supplement what a textbook does by presenting
to the reader something he cannot easily obtain elsewhere:
excerpts from the words of mathematical pioneers themselves.
Most people with any pretense to an education have heard
the names of Euclid, Descartes, and Russell, but few have
read their works. With this little book we hope to close that
gap and enable a reader not merely to read about these men
and to be told that they are famous, but also to read their
works and to judge for himself why and whether they are justly
famous.
Ideally, these nine selections can and should be read without
need of further explanation from anyone else. If there are any
readers who would like to attempt reading only the nine selections
(Part I of each chapter) without the commentaries
(Part II of each chapter), they should certainly try to do so.
The task is by no means impossible; and what may be lost in
time is probably more than outweighed by the added pleasure
as well as the deeper understanding that such a reader will
carry away with him.
However, the majority of readers will probably not want
to undertake the somewhat arduous task of proceeding without
any help. For them, we have provided the commentaries
in Part II of each chapter. These commentaries are meant to
supplement but not to replace the reader’s own thought about
what he has read. In these portions of each chapter, we point
out what are the highlights of the preceding selection, what
are some of the difliculties, and what additional steps should
be taken in order to understand what the author is driving at.
We also provide some very brief biographical remarks about
the authors and, where necessary, supply the historical background
for the book under discussion. Furthermore, we occasionally
go beyond what the author tells us in his work, and
indicate the significance of the work for other fields and future
developments.
Just as the nine selections give us merely a sampling of
mathematical thought during more than 2000 years, so the
commentaries do not by any means exhaust what can be said
about the various selections. Each commentary is supposed to
help the reader understand the preceding selection; it is not
supposed to replace it. Sometimes, we have concentrated on
explaining the dficult parts of the work being considered;
sometimes, we have emphasized something the author has
neglected; at still other times, we extend the author’s thought
beyond its immediate application. But no attempt is or could
be made at examinin g all of the selections in complete detail
and pointing out everything important about them. Such a
task would be impossible and unending. Different commentaries
do different things; in almost every chapter, the author’s
selection contains more than we could discuss in the commentary.
In short, our aim is to help the reader overcome the more
obvious difhculties SO that he can get into the original work
itself. We do this in the hope that the reader will understand
what these mathematicians have to say and that he will enjoy
himself in doing so. Nothing is more fatal to the progress of
a learner in a science than an initial unnecessary discourage
ment. We have tried to save the reader such discouragement
and to stay by his side long enough and sympathetically enough
so that he can learn directly from these great teachers.
Peter Wolf