Abstract

Realist philosophers of mathematics have accounted for the objectivity and robustness of mathematics by recourse to a foundational theory of mathematics that ultimately determines the ontology and truth of mathematics. The methodology for establishing these truths and discovering the ontology was set by the foundational theory. Other traditional philosophers of mathematics, but this time those who are not realists, account for the objectivity of mathematics by fastening on to: an objective account of: epistemology, ontology, truth, epistemology or methodology. One of these has to stay stable. Otherwise, it is traditionally thought, we have a rampant relativism where ‘anything goes’. Pluralism is a relatively new family of positions. The pluralist in mathematics who is pluralist in: epistemology, foundations, methodology, ontology and truth cannot account for the objectivity of mathematics in either the realist or in the other traditional ways. But such a pluralist is not a rampant relativist. In the paper, I look at what it is to be a pluralist in: epistemology, foundations, methodology, ontology and truth. I then give an account of the objectivity and robustness of mathematics in terms of rigour, borrowings, crosschecking and fixtures—all technical terms defined in the paper. This account is an alternative to the realist and traditional accounts of objectivity in mathematics.