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Quotations

Mathematicians are like lovers ... Grant a mathematician the least
principle and he will draw from it a consequence which you must also
grant him, and from this consequence another.
	- Fontenelle

God made the integers. All the rest is the work of man. - Leopold Kronecker
	  [WHO persecuted Georg Cantor all his life for the latter's work on
	  infinity, and eventually Cantor became unstable and died in
	  a mental institution]

Mathematics is the queen of the sciences, and Arithmetic the queen of
mathematics. - Gauss

I regret that it has been necessary for me in this lecture to
administer such a large dose of four-dimensional geometry. I do not
apologize, because I am really not responsible for the fact that
nature in its most fundamental aspect is four-dimensional. Things are
what they are.
	- A.N. Whitehead (The concept of nature, 1920)

There is no royal road to geometry.
	- Menaechmus, to Alexander the Great

For sheer manipulative ability in tangled algebra Euler and Jacobi
have had no rival, unless it be the Indian mathematical genius,
Srinivasa Ramanujan, in our own century. - E.T. Bell, p.328

Lives

   - Zeno, Eudoxus, Archimedes
   - Descartes
   - Fermat
   - Pascal
   - Newton
   - Leibniz
   - The Bernoullis
   - Euler
   - Lagrange
   - Laplace
   - Monge, Fourier
   - Poncelet
   - Gauss
   - Cauchy
   - Lobatchewsky
   - Abel
   - Jacobi
   - Hamilton
   - Galois : the book's most famous chapter, on Galois, is noted for its
       fanciful and often wholly inaccurate account of the events surrounding
       Galois's death in a duel at the age of twenty
   - Sylvester, Cayley
   - Weierstrass, Sonja Kowalewski [sic]
   - Boole
   - Hermite
   - Kronecker
   - Riemann
   - Kummer, Dedekind
   - Poincaré
   - Cantor

Author bio

Eric Temple Bell (1883-1960) b. Aberdeen, Scotland, was Professor of
Mathematics at the California Institute of Technology.
President of the Mathematical Association of America, a
former Vice President of the American Mathematical Society and of the
American Association for the Advancement of Science. He was on the editorial
staffs of the Transactions of the American Mathematical Society, the American
Journal of Mathematics, and the Journal of the Philosophy of Science. He
belonged to The American Mathematical Society, the Mathematical Association
of America, the Circolo Matematico di Palermo, the Calcutta Mathematical
Society, Sigma Xi, and Phi Beta Kappa, and was a member of the National
Academy of Sciences of the United States. He won the Bôcher Prize of the
American Mathematical Society for his research work. His twelve published
books include The Purple Sapphire (1924), Algebraic Arithmetic (1927),
Debunking Science, and Queen of the Sciences (1931), Numerology (1933), and
The Search for Truth (1934).

Excerpt: CHAPTER ONE: Introduction

This section is headed Introduction rather than Preface (which it really is)
in the hope of decoying habitual preface-skippers into reading -- for their
own comfort -- at least the following paragraphs down to the first row of
stars before going on to meet some of the great mathematicians. I should like
to emphasize first that this book is not intended, in any sense, to be a
history of mathematics, or any section of such a history.

The lives of mathematicians presented here are addressed to the general
reader and to others who may wish to see what sort of human beings the men
were who created modern mathematics. Our object is to lead up to some of the
dominating ideas governing vast tracts of mathematics as it exists today and
to do this through the lives of the men responsible for those ideas.

Two criteria have been applied in selecting names for inclusion: the
importance for modern mathematics of a man's work; the human appeal of the
man's life and character. Some qualify under both heads, for example Pascal,
Abel, and Galois; others, like Gauss and Cayley, chiefly under the first,
although both had interesting lives. When these criteria clash or overlap in
the case of several claimants to remembrance for a particular advance, the
second has been given precedence as we are primarily interested here in
mathematicians as human beings.

Of recent years there has been a tremendous surge of general interest in
science, particularly physical science, and its bearing on our rapidly
changing philosophical outlook on the universe. Numerous excellent accounts
of current advances in science, written in as un-technical language as
possible, have served to lessen the gap between the professional scientist
and those who must make their livings at something other than science. In
many of these expositions, especially those concerned with relativity and the
modern quantum theory, names occur with which the general reader cannot be
expected to be familiar -- Gauss, Cayley, Riemann, and Hermite, for
instance. With a knowledge of who these men were, their part in preparing for
the explosive growth of physical science since 1900, and an appreciation of
their rich personalities, the magnificent achievements of science fall into a
truer perspective and take on a new significance.

The great mathematicians have played a part in the evolution of scientific
and philosophic thought comparable to that of the philosophers and scientists
themselves. To portray the leading features of that part through the lives of
master mathematicians, presented against a background of some of the dominant
problems of their times, is the purpose of the following chapters. The
emphasis is wholly on modern mathematics, that is, on those great and simple
guiding ideas of mathematical thought that are still of vital importance in
living, creative science and mathematics.

It must not be imagined that the sole function of mathematics -- "the
handmaiden of the sciences" -- is to serve science. Mathematics has also been
called "the Queen of the Sciences." If occasionally the Queen has seemed to
beg from the sciences she has been a very proud sort of beggar, neither
asking nor accepting favors from any of her more affluent sister
sciences. What she gets she pays for. Mathematics has a light and wisdom of
its own, above any possible application to science, and it will richly reward
any intelligent human being to catch a glimpse of what mathematics means to
itself. This is not the old doctrine of art for art's sake; it is art for
humanity's sake. After all, the whole purpose of science is not technology --
God knows we have gadgets enough already; science also explores depths of a
universe that will never, by any stretch of the imagination, be visited by
human beings or affect our material existence. So we shall attend also to
some of the things which the great mathematicians have considered worthy of
loving understanding for their intrinsic beauty.

Plato is said to have inscribed "Let no man ignorant of geometry enter here"
above the entrance to his Academy. No similar warning need be posted here,
but a word of advice may save some overconscientious reader unnecessary
anguish. The gist of the story is in the lives and personalities of the
creators of modern mathematics, not in the handful of formulas and diagrams
scattered through the text. The basic ideas of modern mathematics, from which
the whole vast and intricate complexity has been woven by thousands of
workers, are simple, of boundless scope, and well within the understanding of
any human being with normal intelligence. Lagrange (whom we shall meet later)
believed that a mathematician has not thoroughly understood his own work till
he has made it so clear that he can go out and explain it effectively to the
first man he meets on the street.

This of course is an ideal and not always attainable. But it may be recalled
that only a few years before Lagrange said this the Newtonian "law" of
gravitation was an incomprehensible mystery to even highly educated
persons. Yesterday the Newtonian "law" was a commonplace which every educated
person accepted as simple and true; today Einstein's relativistic theory of
gravitation is where Newton's "law" was in the early decades of the
eighteenth century; to-morrow or the day after Einstein's theory will seem as
"natural" as Newton's "law" seemed yesterday. With the help of time
Lagrange's ideal is not unattainable.

Another great French mathematician, conscious of his own difficulties no less
than his readers', counselled the conscientious not to linger too long over
anything hard but to "Go on, and faith will come to you." In brief, if
occasionally a formula, a diagram, or a paragraph seems too technical, skip
it. There is ample in what remains.

Students of mathematics are familiar with the phenomenon of "slow
development," or subconscious assimilation: the first time something new is
studied the details seem too numerous and hopelessly confused, and no
coherent impression of the whole is left on the mind. Then, on returning
after a rest, it is found that everything has fallen into place with its
proper emphasis -- like the development of a photographic film. The majority
of those who attack analytic geometry seriously for the first time experience
something of the sort. The calculus on the other hand, with its aims clearly
stated from the beginning, is usually grasped quickly. Even professional
mathematicians often skim the work of others to gain a broad, comprehensive
view of the whole before concentrating on the details of interest to
them. Skipping is not a vice, as some of us were told by our puritan
teachers, but a virtue of common sense.

As to the amount of mathematical knowledge necessary to understand everything
that some will wisely skip, I believe it may be said honestly that a high
school course in mathematics is sufficient. Matters far beyond such a course
are frequently mentioned, but wherever they are, enough description has been
given to enable anyone with high school mathematics to follow. For some of
the most important ideas discussed in connection with their originators --
groups, space of many dimensions, non-Euclidean geometry, and symbolic logic,
for example -- less than a high school course is ample for an understanding
of the basic concepts. All that is needed is interest and an undistracted
head. Assimilation of some of these invigorating ideas of modern mathematical
thought will be found as refreshing as a drink of cold water on a hot day and
as inspiring as any art.

To facilitate the reading, important definitions have been repeated where
necessary, and frequent references to earlier chapters have been included
from time to time.

The chapters need not be read consecutively. In fact, those with a
speculative or philosophical turn of mind may prefer to read the last chapter
first. With a few trivial displacements to fit the social background the
chapters follow the chronological order.

It would be impossible to describe all the work of even the least prolific of
the men considered, nor would it be profitable in an account for the general
reader to attempt to do so. Moreover, much of the work of even the greater
mathematicians of the past is now of only historical interest, having been
included in more general points of view. Accordingly only some of the
conspicuously new things each man did are described, and these have been
selected for their originality and importance in modern thought.

Of the topics selected for description we may mention the following (among
others) as likely to interest the general reader: the modern doctrine of the
infinite (chapters 2, 29); the origin of mathematical probability (chapter
5); the concept and importance of a group (chapter 15); the meanings of
invariance (chapter 21); non-Euclidean geometry (chapter 16 and part of 14);
the origin of the mathematics of general relativity (last part of chapter
26); properties of the common whole numbers (chapter 4), and their modern
generalization (chapter 25); the meaning and usefulness of so-called
imaginary numbers -- like [underroot-1] (chapters 14, 19); symbolic reasoning
(chapter 23). But anyone who wishes to get a glimpse of the power of the
mathematical method, especially as applied to science, will be repaid by
seeing what the calculus is about (chapters 2, 6).

Modern mathematics began with two great advances, analytic geometry and the
calculus. The former took definite shape in 1637, the latter about 1666,
although it did not become public property till a decade later. Though the
idea behind it all is childishly simple, yet the method of analytic geometry
is so powerful that very ordinary boys of seventeen can use it to prove
results which would have baffled the greatest of the Greek geometers --
Euclid, Archimedes, and Apollonius. The man, Descartes, who finally
crystallized this great method had a particularly full and interesting life.

Bell's gifts as a raconteur have earned him some critics among professional
historians, who question his accuracy when it comes to detail. - Did Galois
really feverishly write down all of his mathematical achievements during the
night before the duel that he knew was going to end his life at 20? - Is the
account of Descartes' travails at Queen Christina's court in Stockholm
accurate in every respect? - When Gauss was 9 or 10 years old, his math
teacher wanted some quiet, so the class was assigned the task of adding all
numbers between 1 and 100. Gauss finished it in seconds, while his classmates
toiled away. According to Bell, Gauss proudly told the story all his life: he
had been the only one to come up with the correct answer. - True or not? Who
cares, the point is that the story is credible and says something about
Gauss' natural aptitude for mathematics.