Mathematical Progress As Increased Scope

Abstract

Well-chosen languages contribute to problem-solving success in the history of mathematics. Innovations in notation may not constitute progress on their own for philosophers who reject formalism. Yet successful languages are important concomitants of progress insofar as they enable mathematicians to state claims more broadly and recognize obscure relationships between different branches of mathematics. Platonists recognize the importance of Poincare’s ’happy innovations of language’ as a means whereby ever more mathematical reality is revealed. However, the distinction between what mathematics is about and the formal means used to study mathematics can rarely be made precise outside of isolated historical contexts. Understanding mathematical progress as increased scope is thus an alternative to Platonism’s “mathematical progress as genuine discovery” and formalism’s “mathematical progress as clever invention.”