What Mathematicians Do: Mathematics as Process and Creative Rationality

Mathematics can be viewed through complementary lenses as structure or as process. The latter point of view which is taken in this chapter (and this volume) emphasizes “doing mathematics” and so is tied to learning and creativity. Mathematics is not just any process but a rational one and this chapter investigates what is meant by reason in mathematics. A look at the way mathematics is actually done leads to an expansion of the usual understanding of reason by integrating of intuition and creativity into the rational process. This leads us to take another look at what we mean by objectivity in mathematics. We identify weak and strong varieties of objectivity (analogous to weak and strong AI) and claim that mathematics is objective in the weak sense but not the strong. As a consequence of this and the central place we give to process, mathematics is inescapably tied to human culture and thus never absolutely objective, true, or eternal