A Substitutional Framework for Arithmetical Validity

A platonist in mathematics believes that arithmetic has a subject matter, i.e., that the statements of arithmetic are about certain objects – the natural numbers. For a platonist, the language of (first-order) arithmetic La is referential and he is licensed to speak of true and false sentences of La and to endorse Tarski’s analysis of truth. It follows from this Tarskian analysis plus the fact that every natural number is denoted by some closed term of La (a numeral, if one insists on canonicity) that the truth values of arithmetical sentences are determined by the truth values of its atomic sentences. Consider now a philosopher who, while not a platonist in any sense, broadly accepts the results of mathematics – however tentatively – and is persuaded that the truths of arithmetic are determined by the truth values of its atomic sentences (whatever may be his reformulation of the notion of arithmetical truth). This article may be viewed as an attempt to frame a position for such a non-platonist philosopher of a non-revisionist bent. It is well known that certain atomic sentences of arithmetic have a persuasive rendering in terms of schemata of formulas of first-order languages with equality. This rendering is specially persuasive insofar as we focus on the cardinal role of numbers (and leave their ordinal role aside). For instance, the sentence 7+5 =12 can be rendered as