The later Wittgenstein’s guide to contradictions

Abstract

This paper portrays the later Wittgenstein’s conception of contradictions and his therapeutic approach to them. I will focus on and give relevance to the Lectures on the Foundations of Mathematics (LFM 1976), plus the Remarks on the Foundations of Mathematics (RFM 2001). First, I will explain why Wittgenstein’s attitude towards contradictions is rooted in: (a) a rejection of the debate about realism and anti-realism in mathematics; and (b) Wittgenstein’s endorsement of logical pluralism. Then, I will explain Wittgenstein’s therapeutic approach towards contradictions, and why it means that a contradiction is not a problem for logic and mathematics. Rather, contradictions are problematic when we do not know what to infer from them. Once a meaning is established through a new rule of inference, the contradiction becomes a usable expression like many others in our inferential apparatus. Thus, the apparent problem is dissolved. Finally, I will take three examples of dissolved contradictions from Wittgenstein to clarify further his notion. I will conclude considering why his position on contradictions led him to clash with Alan Turing, and whether the latter was convinced by the Wittgensteinian proposal.