Wittgenstein On Surveyability of Proofs

Abstract

Philosophers of mathematics are often divided between 'mainstream' and 'maverick'. Ludwig Wittgenstein could hardly be classified as 'mainstream'. He had in common with other 'mavericks', however, a critical stance towards the value of the 'mainstream' philosophy of mathematics of his days, mostly but not only as he knew it from the logicist tradition of Gottlob Frege, Bertrand Russell, Frank Ramsey, and Rudolf Carnap. Wittgenstein's remarks on the 'Übersichtlichkeit' or 'surveyability' of proofs were read as forming part of an argument in support of a radical 'anti-realist' account of mathematics, which coincides with 'strict finitism' in the foundations of mathematics. This article argues that Wittgenstein's 'surveyability argument' is best understood not as an argument concerning the length of proofs but as involving the recognition that formal proofs possess a non-eliminable visual element, neglected by Russell. It also discusses the consequences that Wittgenstein drew from his argument, especially as they relate to the larger issue of the cogency of mathematical 'explicativism'.