Wittgenstein on equinumerosity and surveyability

This paper aims to connect two of Wittgenstein's arguments against Logicism. The 'modality argument' is directed at the Frege/Russell-definition of numbers in terms of one-one correlations. According to this argument, it is only when the Fs and Gs are few in number that one can know that they can be one-one correlated without knowing their numbers. Wittgenstein's 'surveyability argument' purports to show that only a limited portion of arithmetic can actually be proven within Principia Mathematica. For proof-constructions within this system quickly become unsurveyable and thereby loose their cogency. As we shall argue, the role of visualisation in proofs plays a fundamental role in both arguments.