I'm just wondering about the spectrum of abstract thinking among the math lovers here. How pure do you like math? Do you insist on visualizing your math problems to be more than a collection of sets and mappings? I couldn't phrase everything I wanted to say in my 5 choices (limited to only 100 characters). Physical application is just one of the major measurements I for abstractness of a math topic. But of course, there are others. Let's say abstractness is measured by how applicable it is directly to real-life situations, how symbolic it is, how difficult it is to visulize, how untangible it is (whatever that means).