ORIGINAL ARTICLE The Role of Axioms in Mathematics Kenny Easwaran Received: 17 November 2006 / Accepted: 28 March 2007 / Published online: 21 March 2008 Ó Springer Science+Business Media B.V. 2008 Abstract To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide variety of philosophical positions. Once the axioms are generally accepted, math- ematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of math- ematics, including with regards to the controversial ‘‘large cardinal axioms’’ Maddy would like to support. 1 Introduction An important contemporary debate (going back to (Go ¨del 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, K. Easwaran (&) Group in Logic and the Methodology of Science, University of California, 314 Moses Hall, Berkeley, CA 94720, USA e-mail: easwaran@berkeley.edu 123 Erkenn (2008) 68:381–391 DOI 10.1007/s10670-008-9106-1