BOOK REVIEW
Reading the Masters
Victor Katz
The Calculus Gallery: Masterpieces from Newton to Lebesgue.
William Dunham. xvi + 236 pp. Princeton University Press, 2005. $29.95.
Musings of the Masters: An Anthology of Mathematical
Reflections. Edited by Raymond G. Ayoub. xvi + 277 pp. The
Mathematical Association of America, 2004. $47.95 ($37.95 for members).
The history of calculus is fairly well known and is competently
described in many places. The prehistory of calculus begins with
Archimedes in the third century B.C. and continues through the works
of some Islamic mathematicians in the medieval period. The first
two-thirds of the 17th century saw the development of integration by
such mathematicians as Johannes Kepler, Bonaventura Cavalieri,
Evangelista Torricelli and Pierre de Fermat; during the same period,
Fermat, René Descartes and others devised approaches to
finding maxima, minima and tangents. Toward the end of the century,
Isaac Newton and Gottfried Leibniz wove together many of these
earlier ideas into "the calculus," although they were not
able to put their creation on a rigorous footing. The 18th century
saw great development of the subject and its applications to more
and more areas of science. But it was only in the 19th century that
Augustin-Louis Cauchy, Karl Weierstrass and others gave calculus a
foundation as secure as the paradigmatic mathematical topic of geometry.
Although this outline is familiar, the mathematical details are less
well known—and quite fascinating. Yet they are often skipped
over in general histories. William Dunham, however, has remedied
this situation in his brilliant book The Calculus Gallery,
at least for the period beginning with Newton and Leibniz. Dunham
picks out important results in calculus from the works of 13
mathematicians, sets them in the context of their times and explores
the original proofs. Although some of the ideas he discusses are
rather difficult, he manages to make them accessible to anyone with
a background equivalent to that of a senior in college majoring in
mathematics. As Dunham points out, these ideas are some of the
masterworks of analysis, and anyone studying mathematics should be
as familiar with these as someone studying art is with the paintings
of Michelangelo and Renoir. I predict that Dunham's book will itself
come to be considered a masterpiece in its field.
Each chapter of The Calculus Gallery is an outstanding
piece of exposition, often based on a talk Dunham has given. I will
discuss just four chapters. In the one on Leonhard Euler
(1707-1783), Dunham not only analyzes Euler's differential approach
to the calculation of the derivative of the sine function but also
shows how Euler found integrals of "bizarre" functions as
well as sums of several interesting infinite series. In
particular, Dunham demonstrates why the sum of the
series of the reciprocals of the integral squares is
π^{2}/6, showing that Euler found this sum, along with
numerous others, as a corollary to a general result on the sums of
reciprocals of the roots of polynomial functions or functions
represented by power series.
In the chapter on Bernhard Riemann (1826-1866), we see one of the
earliest examples of a "pathological" function, a function
with properties that surprised many mathematicians of the day and
led, along with other such functions, to the necessity for a
reevaluation of intuition in analysis. Riemann's example was of a
function that had infinitely many discontinuities on a finite
interval and yet was Riemann-integrable.
The chapter on Karl Weierstrass (1815-1897), in addition to
discussing his influence on the general development of rigor in
analysis, presents one of his own creations, a pathological function
that is continuous everywhere but differentiable nowhere. Although
Weierstrass's proof that his function satisfies those conditions is
long and tricky, Dunham succeeds in making the explanation
understandable to those with the patience to go through each step.
Finally, in the chapter on René Baire (1874-1932), we get a
marvelous discussion of the famous Baire category theorem. This
result always appears in graduate courses on analysis, but for many
of us, there was little besides the strange name by which to
remember it. Dunham changes all that by presenting in great detail
the context of the theorem. He then carefully states and proves it,
using Baire's original proof. Finally Dunham demonstrates that the
theorem enables one to prove easily many of the results that he had
presented in the chapters on earlier mathematicians. For example,
one corollary of the theorem is Georg Cantor's result that, given
any sequence of distinct real numbers and any interval, there is a
point in the interval not a member of the sequence. This immediately
shows, as Cantor demonstrated, that the set of real numbers has
cardinality larger than that of the natural numbers.
Although The Calculus Gallery requires some background in
advanced calculus for a complete understanding, the other book under
review, Musings of the Masters, has few explicit
mathematical prerequisites. The editor of this anthology, Raymond
Ayoub, has collected 17 essays, originally written between 1869 and
1978, by prominent mathematicians, each reflecting the author's
views on what the editor calls the "humanistic" side of
mathematics. Most of the essays are, in fact, the transcripts of
lectures given by these mathematicians to general audiences at such
occasions as the International Congress of Mathematicians.
Given that the book's title includes the word "musings,"
it is not surprising that these essays form a varied lot, dealing
with topics as diverse as the existence of God and Goethe's opinions
about mathematics. To help us put things in context, each piece is
prefaced by both a biographical sketch of the author and a short
summary by Ayoub. The book contains too many essays to discuss them
all here individually, but let me offer my own brief musings about
some of them.
The oldest essay in the collection is James Joseph Sylvester's
presidential address to the British Association in 1869. This noted
English mathematician criticizes remarks of the biologist T. H.
Huxley to the effect that mathematicians spend their time making
"subtle" deductions from a few simple,
"self-evident" propositions, and that mathematics
"knows nothing" of observation, experiment, induction or
causation. Sylvester then brings to bear his immense erudition to
show that, in fact, Huxley's ideas are totally opposite to actual
practice: Mathematicians have always observed and experimented. From
this Sylvester concludes that the study of Euclid in English schools
should be "honorably shelved" and replaced by more
"living" topics that would better stimulate the minds of
students. Interestingly enough, his wishes with regard to Euclid
were achieved. But more than a century later it seems doubtful that,
in England or elsewhere, the replacements do a better job of
providing stimulation.
G. H. Hardy's 1929 lecture on "Mathematical Proof" deals
with the philosophies of mathematics that were then current. Hardy
explains why he is not convinced that any of those philosophies
could be acceptable to the vast majority of working mathematicians,
including himself. As he points out, mathematicians believe that
when they have proved a theorem, they really "know"
something; the philosophies he discusses apparently are too
restrictive in confirming such beliefs.
The only contribution by a woman in this volume is a lecture by Mary
Cartwright on "Mathematics and Thinking Mathematically,"
given at Goucher College in 1969. In her lecture, Cartwright
attempts to distinguish abstract mathematical thinking from the
applications of mathematics in the real world. But, as she points
out, some of the best "abstract" mathematicians formulate
ideas in terms of that real world. For example, her frequent
collaborator J. E. Littlewood worked on antiaircraft gunfire in the
First World War and thereafter often translated more abstract
problems into the language of "trajectories."
Several of the essays deal, directly or indirectly, with the history
of mathematics. André Weil, in a talk given at the
International Congress of Mathematicians (ICM) in 1978, presents his
thoughts on the reasons for studying the history of mathematics. My
own opinion is that too often he looks at ideas from a past era
through a modern lens, neglecting the context in which the ideas
arose. Raymond Wilder, in an ICM talk from 1950, introduces us to
his views on the cultural bases of mathematics, ideas that
culminated in a book published in 1974. Finally, André
Lichnerowicz, in a lecture from 1955, discusses the meaning of being
a scientist, both historically and in modern times.
Although many of the essays in this collection will stimulate
thought and discussion, it would have been no great loss if some of
these "musings" had remained unpublished. Nevertheless,
the collection as a whole will prove valuable in bringing its
readers the thoughts of well-known research mathematicians on topics
outside their areas of specialty.