Wittgenstein, Peirce, and paradoxes of mathematical proof

Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on the rule-following skepticism. We argue that his intuitions rather reflect resistance to treating meaning as fixed content, and are better understood in the light of C.S. Peirce's distinction between corollarial and theorematic proofs. We show how Peirce's insight that "all necessary reasoning is diagrammatic", vindicated in modern epistemic logic and semantic information theory, helps explain the paradoxical ability of deduction to generate new knowledge and meaning.