Originating from interviews with mathematics colleagues, written accounts of mathematicians engaging with mathematics, and Wes's reflections on his own mathematical work, we describe a process that we call mathematical foresight: the imagining of a resolution to a mathematical situation and a path to that resolution. In a sense, mathematical foresight is the process of imagining a not-yet-experienced mathematical event--the solution to a problem solving scenario, or the creation of a mathematical model for a biological system, for examples that could occur in the future. This future thinking process guides the mathematician's present mathematical activity, motivates them, and bolsters persistence.