I know nothing of abstract algebra. In fact I don't even know what it is :). I do however view mathematicians in the same way I see the programming languages crowd: they like to play with these ideas and extend them for their own sake. The solve these puzzles and show that this principle is true, or that group of formulae can be generalized like so. It's up to the rest of us to notice when a particular problem fits some solution they have provided. Another part of this is: math is like alpha, and computer programming is blub, at least according to the math guys.