One thing that's bothered me ever since I heard about complexity theory is why there is no explanation given as to why polynomial and exponential complexities are given such special status. It's just taken for granted that polynomials and exponential are the 2 classes you care about... with no explanation as to why the rest aren't interesting.
EDIT: Edited to clarify, sorry the question was unclear. Thanks for all the replies, but yes, I know the basics :-) I'm not asking about why polynomial complexities themselves are special. Rather, I was trying to ask why we only ever talk about polynomials and exponentials. The only thing that's ever justified is polynomial-time reducibility within a class (which also inherently defines the P class itself). That part is fine. But then once that's established, people suddenly start talking about exponentials. Sorry, but what just happened? There are an infinite number of complexities in between to consider! Why should I suddenly not care about everything in between? Why am I not taught a single algorithm whose complexity is between these? is my question.
EDIT: Edited to clarify, sorry the question was unclear. Thanks for all the replies, but yes, I know the basics :-) I'm not asking about why polynomial complexities themselves are special. Rather, I was trying to ask why we only ever talk about polynomials and exponentials. The only thing that's ever justified is polynomial-time reducibility within a class (which also inherently defines the P class itself). That part is fine. But then once that's established, people suddenly start talking about exponentials. Sorry, but what just happened? There are an infinite number of complexities in between to consider! Why should I suddenly not care about everything in between? Why am I not taught a single algorithm whose complexity is between these? is my question.