Mathematicians, in the ordinary course of their work, are directed in their thinking toward objects, structures, or states-of-affairs in their various domains of research but they are typically not directed toward the cognitive acts and processes in which the awareness of their research domain is constituted. Being directed toward the normal objects of their research—natural numbers, sets, functions, spaces, groups, and so on, and properties of or relations between such objects—is quite different from being directed toward the consciousness of such objects. If we turn to the cognitive acts and processes of mathematicians, however, then we immediately notice a basic feature that is characteristic of many forms of consciousness: intentionality.