# How Mathematics Can Make a Difference

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### Abstract

Standard approaches to counterfactuals in the philosophy of explanation are geared toward causal explanation. We show how to extend the counterfactual theory of explanation to non-causal cases, involving extra-mathematical explanation: the explanation of physical facts (in part) by mathematical facts. Using a structural equation framework, we model impossible perturbations to mathematics and the resulting differences made to physical explananda in two important cases of extra-mathematical explanation. We address some objections to our approach.