On abstract and algorithmic styles in mathematics
Let us consider the paradigm of the Rubik's cube. When you try to solve this particular problem you start with an experimental approach. After that you remark the stability of some patterns and with those you can find an algorithmic method which uses them as steps to the solution. When the problem is solved if you are realy curious and interested you try to find other solving methods. Then getting few different methods you can compare them and find their common point. But to do this, you have to know, for this particular problem, some notions of the theory of finite groups. In fact, if you have already this background you can solve the problem with an abstract method. But it seems for us difficult to directly apply an abstract method, as an heuristic approach, on a new problem (i.e. with no similarities with other known problems). So, for us, an abstract style seems to be a medium for a better and deeper understanding of a solved problem.