Fundamentals of Mathematics represents a new kind of
mathematical publication. While excellent technical treatises have
been written about specialized fields, they provide little help for
the nonspecialist; and other books, some of them semipopular in
nature, give an overview of mathematics while omitting some
necessary details. Fundamentals of Mathematics strikes a unique
balance, presenting an irreproachable treatment of specialized
fields and at the same time providing a very clear view of their
interrelations, a feature of great value to students, instructors,
and those who use mathematics in applied and scientific endeavors.
Moreover, as noted in a review of the German edition in
Mathematical Reviews, the work is "designed to acquaint the
student] with modern viewpoints and developments. The articles are
well illustrated and supplied with references to the literature,
both current and 'classical.'"The outstanding pedagogical quality
of this work was made possible only by the unique method by which
it was written. There are, in general, two authors for each
chapter: one a university researcher, the other a teacher of long
experience in the German educational system. (In a few cases, more
than two authors have collaborated.) And the whole book has been
coordinated in repeated conferences, involving altogether about 150
authors and coordinators.Volume I opens with a section on
mathematical foundations. It covers such topics as axiomatization,
the concept of an algorithm, proofs, the theory of sets, the theory
of relations, Boolean algebra, and antinomies. The closing section,
on the real number system and algebra, takes up natural numbers,
groups, linear algebra, polynomials, rings and ideals, the theory
of numbers, algebraic extensions of a fields, complex numbers and
quaternions, lattices, the theory of structure, and Zorn's
lemma.Volume II begins with eight chapters on the foundations of
geometry, followed by eight others on its analytic treatment. The
latter include discussions of affine and Euclidean geometry,
algebraic geometry, the Erlanger Program and higher geometry, group
theory approaches, differential geometry, convex figures, and
aspects of topology.Volume III, on analysis, covers convergence,
functions, integral and measure, fundamental concepts of
probability theory, alternating differential forms, complex numbers
and variables, points at infinity, ordinary and partial
differential equations, difference equations and definite
integrals, functional analysis, real functions, and analytic number
theory. An important concluding chapter examines "The Changing
Structure of Modern Mathematics."
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