The Role of Axioms in Mathematics

An important contemporary debate (going back to (Gödel, 1964)) in the philosophy of mathematics is whether or not mathematics needs new axioms. This paper is an attempt to show how one might go about answering this question. I argue that the role of axioms is to allow mathematicians to stay away from philosophical debates, and thus focus on their primary goal of proving interesting theorems. I find four tests that a new axiom candidate must pass in order to be accepted. However, seeing whether an axiom passes some of these tests will involve some serious philosophical work axioms only delay the philosophical questions, they don’t dissolve them. Solomon Feferman opens his contribution (Feferman, 1999) to this debate with the observation: The question, “Does mathematics need new axioms?”, is ambiguous in practically every respect.