On mathematical validity and its human origins

This chapter suggests the desirability of a fully integrated philosophy of mathematics education that builds on several distinct, mutually compatible foundational pillars. These pillars have their intellectual bases in different philosophical school of thought. The discussion focuses on one aspect of those epistemological foundations - the interplay between the human origins and uses of mathematics and its objective truth and validity. Various philosophical trends in education have centralized just one of these aspects, often to the extent of denying or dismissing the other. I argue here for their compatibility, maintaining that objective mathematical truth and the fact of culturally situated human mathematical invention should both be guiding teaching and learning, with neither diminished in importance. Several meanings given to the "why" of mathematics are discussed - logical and empirical reasons that underlie mathematical truths and relationships, sociocultural and contextual reasons for developing and teaching mathematics, and the in-the-moment experiences afforded students to motivate their study. Some sources of cultural relativism, historical change, and "fallibility" in mathematics are identified, and the value of "mistakes" in powerful mathematical problem solving is highlighted. My goal is to argue that an intellectually sound philosophy of mathematics education must incorporate all of the aforementioned features of mathematics and its practice, dismissing none.