I am surprised by the twist this discussion took - saying that applications (meaning?) interfers with the learning of abstract (formal?) mathematics. First - I can imagine teaching music through its applications to everyday life (of the student). The Suzuki method teaching tunes (by ear), long before scales and reading music. I am convinced that one reason my son is now a student of music at university, rather than a student of mathematics (or science) is because his music teacher (private) responded to his interest in 'making music' of a popular sort, rather than doing scales or 'good pieces' picked by the teacher. Unfortunately, none of his mathematics or science teachers were as open. (The computer science teacher was more open - hence his science fair project on analyis of sound on a computer - and his current program in Computers and Music). I learned pure mathematics as a student (almost no geometry), and only later turned to applied mathematics (geometry). The idea of learning projective geometry without the underlying statics (and kinematics - if you are careful) of Mobius et al seems absurd after this experience. Why wouldn't we use the essential issues of, say, computer graphics to motivate and illustrate aspects of geometry? These are neither gimics nor past issues. There is a great deal of interest in geometry arising in applied areas such as computer aided geometric design - and a number of unsolved problems, even in plane Euclidean geometry. Many people not trained in mathematics are discovering they NEED to know geometry. Current engineering text LIE about known results because the students know too little geometry to understand the real answers. I think this reality adds to the enjoyment of geometry.
I suspect that many parts of mathematics are learned as abstract pure mathematics, in part because we have forgotten the original 'practical' issue. Pythagoris Theorem was the solution to a practical problem - how do you add two squares using a straight edge and compass. (Given that you trusted geometry more than algebra, this is 'practical'.)
Yes - applications are not enough. They seldom generate the fundamental insights which unify the pieces - and generate new applications. I think the issue is to balance applications (and interest expressed by the students) with theory that will build for later. I teach (currently) at the undergraduate level. In courses such as geometry - I make it part of every assignement that the students write out some questions THEY had when they finished the assignement. Sometimes these are integrated back into the class work. Sometimes they become a private dialog with the student. Sometimes they become the topic of a major project ( near the end of the course). THis does not always work - but it works better that many of my efforts to guess what is releveant to these students (who tend to people wanting to become mathematics teachers!).