This chapter initiates a path beyond orthodoxy by analyzing the practices of physics, rather than idealized theories. It focuses on the roles of mathematics and symmetry principles in extending beyond the limits of Bohrian semantics. Quantum electrodynamics illustrates the interrelation of formal and informal inferences in interpreting correction terms to the Lamb shift and the anomalous magnetic moment of the electron and muon. These calculations show the necessity of including virtual processes in any account of the quantum realm. The standard model of particle physics relies on local gauge invariance and on symmetry principles as postulations not sanctione4d by orthodox quantum mechanics or algebraic quantum field theory. The chapter concludes with an analysis of the distinctive features of quantum systems: superposition, interference, distributed probability, and entanglement.