Based on a well-received course designed for philosophy students, this
book is an informal introduction to mathematical thinking. The work will
be rewarding not only for philosophers concerned with mathematical
questions but also for serious amateur mathematicians with an interest
in the "frontiers" as well as the foundations of mathematics. In what
might be termed a sampler of the discipline, Konrad Jacobs discusses an
unusually wide range of topics, including such items of contemporary
interest as knot theory, optimization theory, and dynamical systems.
Using Euclidean geometry and algebra to introduce the mathematical mode
of thought, the author then turns to recent developments. In the process
he offers what he calls a "Smithsonian of mathematical showpieces": the
five Platonic Solids, the Mbius Strip, the Cantor Discontinuum, the
Peano Curve, Reidemeister's Knot Table, the plane ornaments, Alexander's
Horned Sphere, and Antoine's Necklace. The treatments of geometry and
algebra are followed by a chapter on induction and one on optimization,
game theory, and mathematical economics. The chapter on topology
includes a discussion of topological spaces and continuous mappings,
curves and knots, Euler's polyhedral formula for surfaces, and the
fundamental group. The last chapter deals with dynamics and contains
material on the Game of Life, circle rotation, Smale's "horseshoe, " and
stability and instability, among other topics.
