Informal Proof, Formal Proof, Formalism, and Fictionalism

The fictionalist aims to avoid commitment to mathematical objects by replacing mathematical truth with fictional correctness: truth-in-the-story (of standard mathematics). For an axiomatically stated mathematical theory T, a sentence S is true-in-the-T-story if and only if it follows logically from the axioms of T. The formalist, on the other hand, seeks to avoid commitment to mathematical objects by replacing mathematical truth with formal derivability: a sentence S is true in a mathematical theory T if there is a derivation of S from the axioms of T. For first-order theories, and if the existence of a derivation is understood to mean the in principle possibility of deriving S from T, formalism and fictionalism would seem to coincide. However, in practice the two views typically diverge, with the divergence being due to different attitudes to modality. For fictionalists such as myself who embrace primitive modality, a modal notion of logical consequence allows for sentences in second order arithmetic to have objective ‘truth-in-the-story’ values even if they are not derivable. On the other hand as the most prominent contemporary advocate of formalism, Alan Weir wishes to develop an austere form of formalism without modal commitments, so that mathematical truth is not grounded in the in principle possibility of a derivation, but only in the existence of actual derivations. In this chapter I continue a long-standing debate between myself and Weir over the nominalistic acceptability of primitive modality, arguing that Weir’s account of informal mathematics involves him in modal commitments whether he likes it or not.