In 1939, Paul Dirac observed that “the physicist, in his study of natural phenomena, has two methods of making progress”: experiment and observation, and mathematical reasoning. Although he said, “there is no logical reason why the second method should be possible,” nevertheless it works, and to great effect. The key, Dirac felt, was beauty, leading him to his principle that successive theories of nature are characterized by increasing mathematical beauty. The results of this were rich and included some predictions not confirmed until after Dirac’s death. Nevertheless, the powerful guidance Dirac found in mathematics did sometimes lead him astray, as he rejected the principle of “renormalization,” developed by Feynman, Schwinger, and Tomonaga, to remedy the nonphysical infinities that kept cropping up in Dirac’s equations for quantum electrodynamics. Even as other physicists accepted it, Dirac never did, saying it was “just not sensible mathematics.” Nevertheless, it was powerful physics. The results were field theories of the electroweak and strong forces and the Standard Model of particle physics -- a theory arguably endowed with its own “unreasonable effectiveness.” This session will discuss how beautiful mathematics has (and has not) served theoretical physics from the 1930s to the modern hunt for the Higgs boson, supersymmetry, and whatever else might be found by the Large Hadron Collider and how both empiricism and mathematics will direct the physics of the future.