ABSTRACT
This chapter concerns issues of mathematical philosophy and pedagogy. The past few years have seen attempts on the part of philosophers and mathematicians to reconceptualize and redescribe the mathematical enterprise. Mathematical authors such as Steen and Hoffman have tried to frame a popular notion of mathematics that accurately reflects the nature of contemporary mathematics and that also serves as a basis for a modern pedagogy of mathematics. The ideas expressed here are grounded in some philosophical reflections, but they start off in a practical vein. It deals with issues of ontology and/or epistemology or in more down-to-earth language, what it means to do mathematics. The chapter is grounded in epistemological issues but focuses on issues of instruction, provides descriptions of selected aspects of the author's problem-solving courses. The goal of his problem-solving courses is to deflect inappropriate teacher authority. Through such processes the mathematical community implements mathematical authority with consistency and with accuracy.
