We Hold these Truths to be Self-Evident: but What do we mean by that?

At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are undermined at a crucial point, namely when self-evidence is supported by holistic and even pragmatic considerations. At the beginning of Die Grundlagen der Arithmetik (§2) (1884), Gottlob Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”, noting that “Euclid gives proofs of many things which anyone would concede him without question”. Frege sets himself the task of providing proofs of such basic arithmetic propositions as “every natural number has a successor”, the induction principle, and “1 + 1 = 2”. Frege’s observation was true in Euclid’s day, and it remains true now. We still admire the achievements of Euclid, Archimedes, Cauchy, Weierstrass, Bolzano, Dedekind, Frege, and a host of others on providing rigorous proofs of “many things that formerly passed as self-evident”, as Frege put it (§1). Many of these are propositions that no one in their right mind would doubt—unless it be on skeptical or nominalist grounds (in which case mathematical proof would not settle the issue). Nevertheless, thinkers differ widely on why it is that we prefer proof, and this question goes to the very heart of mathematics. My topic here is closely related to this. It is a commonplace that one cannot provide a nontrivial or noncircular proof of every known proposition. Frege’s observation is that mathematics prefers proof, “where proof is possible”. What about cases where proof is not possible? What is the epistemic status of the axioms, or basic truths (or inference principles), from which other propositions are derived? If we claim to know the theorems, on the basis of the proofs, then surely we must claim to know the axioms (or inference principles)? How? §1. Axioms as definitions: They tell us what we are talking about.1 A relatively recent perspective is that the axioms of a given branch of mathematics serve as an implicit Received: December 5, 2008 1 I do not claim that what is said in this section is definitive, but pursuing the matter further would take us too far afield. For more, see Shapiro (2005). The purpose of this section, and the next two, is to raise and give reason to dismiss certain conceptions of axioms, before we take on the notion of self-evidence. c © 2009 Association for Symbolic Logic 175 doi:10.1017/S175502030909011X