An Episodic History of
Mathematical Culture through Problem Solving
by Steven G. Krantz
September 23, 2006
To Marvin J. Greenberg, an inspiring teacher.
Together with philosophy, mathematics is the oldest academic dis-
cipline known to mankind. Today mathematics is a huge and complex
enterprise, far beyond the ken of any one individual. Those of us who
choose to study the subject can only choose a piece of it, and in the end
must specialize rather drastically in order to make any contribution to
the evolution of ideas.
An important development of twenty-first century life is that mathe-
matical and analytical thinking have permeated all aspects of our world.
We all need to understand the spread of diseases, the likelihood that we
will contract SARS or hepatitis. We all must deal with financial matters.
Finally, we all must deal with computers and databases and the Internet.
Mathematics is an integral part of the theory and the operating systems
that make all these computer systems work. Theoretical mathematics is
used to design automobile bodies, to plan reconstructive surgery proce-
dures, and to analyze prison riots. The modern citizen who is unaware
of mathematical thought is lacking a large part of the equipment of life.
Thus it is worthwhile to have a book that will introduce the student
to some of the genesis of mathematical ideas. While we cannot get into
the nuts and bolts of Andrew Wiles’s solution of Fermat’s Last Theorem,
we can instead describe some of the stream of thought that created the
problem and led to its solution. While we cannot describe all the sophis-
ticated mathematics that goes into the theory behind black holes and
modern cosmology, we can instead indicate some of Bernhard Riemann’s
ideas about the geometry of space. While we cannot describe in spe-
cific detail the mathematical research that professors at the University
of Paris are performi