Is Mathematics Unreasonably Effective?

Many mathematicians, physicists, and philosophers have suggested that the fact that mathematics—an a priori discipline informed substantially by aesthetic considerations—can be applied to natural science is mysterious. This paper sharpens and responds to a challenge to this effect. I argue that the aesthetic considerations used to evaluate and motivate mathematics are much more closely connected with the physical world than one might presume, and (with reference to case studies within Galois theory and probabilistic number theory) I show that they are correlated with generally recognised theoretical virtues, such as explanatory depth, unifying power, fruitfulness, and importance.