The philosophy of mathematics of the last few decades is usually distinguished into mainstream and maverick.1 The mainstream philosophy of mathematics considers mathematics as a static body of knowledge; it is mainly concerned with the question of the justification of mathematical knowledge; it holds that there is an absolutely certain, or at least fairly reliable, foundation for mathematics; it considers mathematical logic as a canon for the philosophy of mathematics; it assumes that a detailed account of mathematical practice would be desirable but not really essential; it generally sets itself within the framework of analytic philosophy. The maverick philosophy of mathematics considers mathematics as a dynamic body of knowledge; it is mainly concerned with the question of the growth of mathematical knowledge, including the dynamics of mathematical discovery; it holds that there is no absolutely certain foundation for mathematics; it considers mathematical logic very useful to show the limitations of the mainstream philosophy of mathematics by means of the limitative results, but inadequate to deal with the question of the growth of mathematical knowledge; it assumes that only a detailed analysis of mathematical practice could lead to a philosophy of mathematics worth its name; it generally sets itself outside the framework of analytic philosophy. The mainstream philosophy of mathematics consists of the three big foundational schools of the first few decades of the twentieth century, namely logicism (Frege, Russell), formalism (Hilbert), intuitionism (Brouwer, Heyting), and the positions which ensued from them in the second half of the twentieth...