The historical development of the gamma and beta functions exemplifies a fundamental epistemological shift in mathematics during the eighteenth and nineteenth centuries—a transformation from computational tools grounded in discrete operations to abstract functions capable of traversing multiple representational domains. This chapter examines how the iterative refinement of these special functions reflects broader mathematical trends toward generalization, abstraction, and “representational flexibility”—the capacity to switch fluidly between different mathematical formulations of the same underlying concept.