
nine year old Malone writes in the introduction to his final volume, The Sage of
Monticello, of the life of a fellow octogenarian: “It has been my great privilege as a
biographer to be intimately associated with this extraordinary man for many years. At the
end of my long journey with him I leave him with regret and salute him with profound
respect.” Thomas Jefferson was indeed an extraordinary man whose interests,
personality, and intellectual genius can be more fully appreciated in the 20th century
because of Malone’s biographical classic.
WILLIAM J. ZIOBRO
Philip J. Davis and Reuben Hersh
The Mathematical Experience (1981)
“Statesmen despise publicists, painters despise art critics, and physiologists, physicists, or
mathematicians have usually similar feelings; there is no scorn more profound, or on the
whole more justifiable, than that of the men who make for the men who explain.
Exposition, criticism, appreciation, is work for second-rate minds.” So wrote G.H. Hardy
at the beginning of A Mathematician’s Apology, one of the most beautiful books in or out
of mathematics, and a book in which a first-rate mind turned his attention to exposition,
criticism, and appreciation of mathematics. Since that time, fortunately, other first-rate
minds have been drawn in the same direction, exposing many different views of the
nature of mathematics. The latest effort along these lines, by Philip J. Davis and Reuben
Hersh, is perhaps the most ambitious and best of all in describing what mathematicians
really do. Let me state plainly at the outset that every student of mathematics will find
this book valuable, and that Davis and Hersh, who write as one, doubly deserve our
gratitude.
What is mathematics, and what manner of spirit animates its study? That, of
course, is a philosophical question. One of many favors done by Davis and Hersh is to
clear the air a bit by pointing out that the so-called philosophies of logicism, formalism,
and intuitionism are not and never have been philosophies of mathematics. Their focus,
of course has always been on the foundations of mathematics, and they became known as
“philosophies” only because there was a period in the early part of this century when
almost all work in the philosophy of mathematics was work in foundations. Despite its
great importance, the study of foundations is only one of a multitude of interesting
aspects of mathematics.
In fact, the diversity of interesting aspects is bewildering, as we find ourselves
saying that mathematics is like X, like Y, and like Z, when X, Y, and Z themselves are
quite dissimilar. Davis and Hersh give us a number of arguments, each well done, from
the totality of which we must conclude that mathematics is like an ideology, a religion, or
an art form, and is thus a humanistic study, “one of the humanities”; and yet mathematics
has a science-like quality in that “its conclusions are compelling, like the conclusions of
natural science.”
“Mathematics, being a human activity,” they say, “. . . profits greatly from
individual genius, but thrives only with the tacit approval of the wider community. As a
great art form, it is humanistic; it is scientific-technological in its applications.”
Mathematics is thus caught in the struggle between the individual and society as well as
the struggle between the arts and sciences. In the arts-science tension there is nothing
Return to Index 66