Relevant Arithmetic

Relevant arithmetic is the project of reconstructing number theory within relevance logic rather than classical logic. The central challenge is that classical Peano arithmetic relies on structural rules — especially weakening and contraction — that relevance logics reject. Robert K. Meyer proved that relevant Peano arithmetic (R#) is consistent relative to the consistency of the base relevance logic R, but also showed that it is weaker than classical arithmetic in ways that reveal the logical sensitivity of mathematical content.

Relevant arithmetic demonstrates that the choice of underlying logic is not merely a matter of inference rules but a constraint on what mathematics can express. It connects the philosophical demand for relevance to the foundational structure of number theory, and raises the question of whether other mathematical domains — analysis, algebra, geometry — have similarly been shaped by unnoticed classical assumptions.\n\nThe extension of relevant methods to analysis and geometry remains largely unexplored.