Mathematics: Its Spirit and Evolution
John R. Durbin
This book is out of print. The following review appeared in the American Mathematical
Monthly, Vol. 80, No. 9, Nov. 1973.
Mathematics: Its Spirit and Evolution. By John R. Durbin. Allyn and Bacon, Inc.,
Boston Massachusetts, 1973. xii + 321 pp. $10.95.
Offering refreshing departures from the usual ahistorical survey, this college
textbook is John R. Durbin’s answer to “the question of how the spirit of mathematics
can best be conveyed to those with limited background in the subject” (p. ix). Through
his choice and arrangement of material from the mathematical corpus Prof. Durbin
clearly and concisely imparts to the reader a sense of the history, spirit and unity of
mathematics.
Prof. Durbin’s approach is twofold: the first two-thirds of the book (excluding the
first chapter which briefly develops the concepts of rational, irrational and real number)
presents in-depth investigations of only a few important mathematical topics; the
remaining one-third discusses briefly and often interrelates many basic concepts in order
to provide “a view of some general characteristics of modern mathematics” (p. xi).
Chapters II through V, respectively, deal in detail with non-Euclidean geometry,
probability, cardinal numbers, and groups.
Of these topics, two — non-Euclidean geometry and groups — are treated in very
great thoroughness and depth for an introductory liberal arts mathematics textbook. The
discussion on non-Euclidean geometry begins with a probing treatment of Euclid’s
Elements which includes a complete listing from Heath’s edition of the 23 definitions, 5
postulates, 5 common notions, and 48 propositions of Book I. Most important, however,
is the analysis of the logical status of Euclid’s fifth postulate and of the non-Euclidean
geometries which result by employing alternatives to it, because, as Prof. Durbin puts it,
“no topic is better suited for showing what mathematics is and what it is not” (p. 38).
Undoubtedly Prof. Durbin has in mind the fact that the subject matter of non-Euclidean
geometry forces the student to divest himself of his Euclidean physical intuitions of space
in order to appreciate as a valid mathematical system one which runs counter to those
intuitions. In any case, 16 theorems of hyperbolic geometry are stated and proved,
including several on Saccheri quadrilaterals and the one which states that the sum of the
angles of any triangle is less than two right angles. The chapter on groups provides a
detailed analysis of the use of the group concept in analyzing symmetries, deals with
isomorphisms and permutation groups, and concludes with the statement and proof of
both Lagrange’s and Cayley’s Theorems. These two in-depth explorations of deductive
mathematical systems, one geometric, the other algebraic, forcefully convey to the
student the axiomatic nature of mathematics.
Chapter VI, which comprises the final one-third of the book, is designed
differently. Here many mathematical concepts and ideas are treated briefly, frequently
interrelated with each other, and all shown to be part of a structure and technique that
fairly can be called elementary mathematical analysis. An historical overview is first
presented, then the three famous problems of Greek mathematics are posed, conic
sections are introduced, the Eudoxian method of exhaustion discussed, the sine, cosine,
logarithmic and exponential functions are defined, and the nature and power of analytic
geometry illustrated, and so on. Without a further detailed listing of specific topics,
suffice it to say that enough conceptual mathematics is presented lucidly so that when the
fundamental theorem of arithmetic, the fundamental theorem of algebra, and the
fundamental theorem of the calculus appear, each in an appropriate context, they should
readily be understandable to the student. Chapter VI concludes with a discussion of the
distribution of primes, including the role of the prime number theorem, a brief exposition
of the zeta function and the Riemann Hypothesis concerning its roots. This chapter, with
its depth of treatment its many interrelations of concepts and reference to ideas
previously considered, well illustrates the unity and power — even the beauty — of the
mathematical enterprise.
This is a no-nonsense book for the general education mathematics student, but
one which could be adapted for a variety of teaching situations, because, as the author
points out, “the chapters, after the first, are nearly independent of one another” (p. xi),
thus allowing for rearrangements or selective omissions. The gradation of the problems
as to their difficulty is also conducive to this end. It should be noted that the book is
remarkably free of printing errors. More important, however, is the interweaving of an
accurate and sensitive historical commentary throughout the book which justifies the
word “Evolution” in its title. In sum, John R. Durbin’s book, a captivating blend of rigor
and vigor, of mathematics and mathematical history, is one that I recommend for
consideration by those who teach liberal arts mathematics.
Randal Longcore, Atkinson College; York University, Toronto