There is a common back-of-the-mind idea, prompted by the counterintuitiveness of quantum phenomena, that quantum states may represent ``mere possibilities'' in some vague sense. Yet, this idea reflects itself neither in the quantum formalism nor in the way we use it. In this paper, I explore what it entails to take this idea seriously.I begin by showing that already axiomatic probability fails to formally distinguish between possibilities and outcomes, even though it is conceptually supposed to be a unit measure over possibilities. I propose an axiomatic enrichment which introduces the distinction into Kolmogorov's axiomatization and then demonstrate how this provides a more faithful model of reality. Next, I explore an analogous mathematical modification of the standard quantum formalism in which Hilbert Space elements are no longer states of physical systems but mere physical possibilities, each linked to one of a set of actual states of systems we can observe if the possibility is actualized. The modified formalism leaves the rules of quantum mechanics the same as before, but makes it possible to distinguish between what Heisenberg called ``potentialities or possibilities'' and ``things and facts'' at the level of the mathematics. I call this the Heisenberg Interpretation, and while it is not testable within the domain of quantum mechanics itself, it does erect a mathematically well-defined separation between quantum and classical physics. If the classical domain is identified with that of general relativity, then the formalism can be tested by checking for the existence of the boundary via proposed experiments like the BMV effect or by measuring the gravity field of an ultra high energy laser. Finally, I argue that physical possibilities are an overlooked conceptual ingredient critical to all of physics.