The applicability of mathematics as an epistemic problem

Abstract

This thesis examines the problem of the applicability of mathematics in empirical science, commonly associated with Wigner’s “unreasonable effectiveness of mathematics.” The central aim is methodological: to determine how the problem of applicability ought to be formulated. The thesis argues for the applicability of mathematics as an epistemic problem. In the first part, some influential formulations are rejected because of their reliance on overly narrow or untenable assumptions about mathematics, empirical science, or their relation, specifically those focused on mathematics-led discovery, the role of aesthetics in mathematics, the metaphysical gap between mathematics and physics, and the use of idealisations. The second part looks at the variety of ways in which mathematics is applied—semantic, descriptive, deductive, non-standard, unificatory, and explanatory—and argues that all but semantic applicability can be understood through deductive applicability, namely the use of mathematical statements as premises in deductions with empirical conclusions. The final part develops an epistemic version of the problem centred on deductive applicability: mathematical statements, generally taken to be empirically indefeasible, transmit justification to conclusions that are defeasible by empirical evidence. The thesis concludes by defending this formulation of the problem and examining how several influential philosophies of mathematics respond to it.