Wittgenstein, Peirce, and paradoxes of mathematical proof

TL;DR

This paper explores Wittgenstein's paradoxes about mathematical proof, interpreting them through Peirce's distinction between types of proofs and emphasizing the role of diagrammatic reasoning in generating new knowledge.

Contribution

It offers a novel interpretation of Wittgenstein's ideas by connecting them with Peirce's proof distinction and modern logic, clarifying the nature of mathematical reasoning.

Findings

Peirce's corollarial and theorematic proofs elucidate Wittgenstein's paradoxes.

Diagrammatic reasoning underpins the generation of new knowledge.

Modern epistemic logic supports Peirce's view on reasoning as diagrammatic.

Abstract

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.