What follows is a report on fragments of studies aimed eventually at a thorough phenomenological investigation of Kurt Gödel's conviction that a form of intuition, not unlike perceptual intuition, plays an essential role in even highly infinitary mathematics. The studies are phenomenological in at least the senses that they attempt to take a microscopically close look at a wide range of purportedly mathematical intuitions, determining on this basis what makes them tick and what they are good for. In them I draw upon and extend Husserl's studies of the nature of such intuition (which he called "categorial intuition'). Indeed, Gödel himself praised them highly, albeit they stand in great need of improvement and development, but decidedly not in the direction of Husserl's late work, Experience and Judgment ².