Some students begin by thinking of mathematics only as a tool required for other work, and then find the tool so interesting that they want to study it more. For such students, mathematics is about solving problems. Such mathematical applications often lead to a secondary major in mathematics or a mathematics minor, something that employers like to see because it assures them of the student's quantitative skills.

Other students are drawn to mathematics for aesthetic reasons, perhaps because they glimpsed the precision and clarity of mathematical thought in their earlier mathematics courses. It is rare in undergraduate study that one can hold a truly important subject in one's mind, feel its theoretical cohesiveness, and see the beauty of it from top to bottom. While such understanding does not come easily, it is possible for an undergraduate in mathematics. And when one grasps the theoretical unity of parts of our subject in that way, mathematics is more like a poem than a tool.

The distinction between applied mathematics and theoretical mathematics has often turned out to be an illusion.  Even the most theoretical parts of mathematics have surprisingly concrete applications.  Number theory, long viewed as the least applicable part of mathematics, is today the foundation for protection of financial records and security of electronic transmissions.  And many of the most applied parts of mathematics have given rise to major parts of pure mathematics.  The study of planetary motion, for example, grew into modern dynamical systems theory.