Chapter 1. Introduction
The present survey deals mostly with rather elementary mathematics,—so elementary in
fact that most of its results and problems are (or at least should be) understandable to under-
graduates. It was written out of the eonvietion that many negleeted aspeets of elementary
geometry deserve a wider dissemination, —because of their inherent beauty and interest, and
for the Inspiration and understanding they can impart to our students and to ourselves.
During the last decade or two» it has become an article of faith with many of us that
researeh in mathematics and its teaching are worthwhile only if they deal with very general
and abstract topics. The elements of algebra, set theory, logic, topology, measure theory,
etc., —that is what advanced undergraduates and starting graduate students are being taught
in good schools; it would be foolish to say that this should be changed. What is unfortunate,
and what cripples most of the students for the rest of their professional lives, is that other
topics and points of view are only rarely presented. The students are made to understand
that researeh in, say, finite groups is "important"—no matter how esoteric the problem in-
vestigated,— while topics in the Euclidean plane, for example, are certainly old-fashioned
since the Euclidean plane is a very special strueture. Many aspeets of geometry fare partic-
ularly badly from this attitude; they are even denied the right to exist unless they fit nice
algebraic patterns. In the most ridiculous and extreme aspeets of such tendencies geometry
is equated with a subdiseipline of linear algebra.
As a counterthrust to that trend I would like to recall just a few of the many historical
instances in which the geometric patterns of thought or points of view led not only to
another theorem or two but were crucial in gaining new and revolutionary insights:
(i) The discovery of irrational numbers;
(ii) The invention of calculus;
(iii) The development of the axiomatic method;
(iv) The emergence of algebraic topology, and of functional analysis.
*) The first version of this survey was presented as an invited address at the Annual Spring Meeting
of the Michigan Section of the Mathematical Association of America held on April 4, 1970, at the Wayne
State University in Detroit, Michigan.
Research sponsored in part by the National Science Foundation through the Science Development
Program grant GU-2648, and by the Office of Naval Research through contraet N00014-67-A-0103 -
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