Showing what physicists are really doing behind the nearly impenetrable cloud of mathematical models they use as research tools, the author argues that the technical details of these complex calculations serve not only as a means to an end, but also reveal key aspects of the physical properties they model. Using two tours de force of modern physics as case studies - proofs that ordinary matter if stable and solutions to the Ising model of a phase transition - the book uncovers the philosophical roots of the mathematical models of these phenomena, showing not only how physicists believe the natural world is structured, but also how they draw those conclusions.