Within contemporary philosophy of mathematics, "classicism" in mathematics is understood as standard mathematical practice as characterized by classical logical reasoning. This practice is usually associated with a "platonist" philosophical standpoint according to which axioms and theorems are viewed to be objective truths about a mind- independent mathematical reality consisting of abstract objects of the sort mathematical discourse appears to quantify over. "Constructivism," on the other hand, covers those minority schools who reject as illegitimate central components of classical practice and seek to develop alternative practice based on constructivist principles. The particular constructivist schools under examination in this project reject as illegitimate the classical conception of objective mathematical truth on which classical logic is based and they reject the ordinary notion of "existence" in context of mathematics. In its place, they substitute principles of constructive object and constructive proof. In this project, Professor Hellman is examining the longstanding opposition between classical mathematical practice and leading constructivist schools, emphasizing the importance of scientific applications of mathematics. At the outset, he is examining the philosophical underpinnings of the constructivist challenge in light of recent "modal-structural interpretations," supporting classicism but avoiding certain problems of standard platonism. He is then examining arguments that abstract mathematics is "indispensable" for science and assessing the scope and force of these arguments, especially in light of recent mathematical-logical work (predicative reductions, reverse mathematics) which seems severely to limit the impact of such arguments. Finally he is bringing to bear such arguments on the classicism-constructivism controversy. The viability of constructivism for scientific purposes is being assessed in light of recently uncovered obstacles to its applicability, both generally, in circumstances of empirical uncertainty, and specially, in the domain of quantum physics.
