I don't think there is any one answer to your question. I think punishment originally came the fact that without it, humans could not live collectively with one another. For example, we live 3,000 years ago as a small tribe. We just learned agriculture and are beginning to witness divisions in labor. However, we really don't like our neighbors because they repeatedly come over and kill us and take our things. So in order to protect ourselves, we decide to band together and form an army. We kill our neighbors together the next they swing by. (We could also decide to visit them if we become impatient, or if we want some of their things.) In my opinion, this the origin of heirarchical society. Driven by the need to protect ourselves from some one else. Militaries (or militias) need leaders to remain effective, so now the need to protect ourselves from others has created a ruling class.

Unfortunately, just having a leader did not solve all of our problems. Ted doesn't work very hard (or has had a very bad run of bad luck), so he decides to steal from some one else in the tribe, Bill. Bill is quite upset to see his possessions being taken by Ted and wants to kill Ted just as he would kill someone from the neighboring tribe. The problem is if every one in the tribe is killing one another, then the tribe won't be unified for long and we will no longer be able to defend against (or pillage) our neighbors.

The only solution is for the ruler to say what Ted did was wrong and punish him. That way Bill is happy (or happier) and chaos does not erupt. So at least in the early days, punishment was meant to re-direct the feelings of revenge in a manner that would not erode the tribe's ability to live together.

As we have moved through time, we learned things like, "well Ted was just down on his luck that day so maybe he doesn't need to be beaten or hung." Hence, we begin to see rehabilitation enter into the frame work of justice.

I have never been a fan of the deterrence argument, i.e., people will not do what Ted does when they see Ted getting punished. Most likely, a criminal mind would say, "Ted should have waited until this time or he should have done that."

There's another prong of the punishment angle which says if you take the criminal off the street, he won't be able to commit other crimes. (recidivism?) I don't know how effective this is, but with the US prison population at record numbers and growing, we should be seeing some empirical results soon, if not already.

I guess what I am saying is, that you can't say it is about this or that, b/c its probably all of the above, except that I don't think the deterrence argument holds water. Crime results from either anger, unmet needs or irrationality/insanity. Someone operating under any of these probably won't be deterred by the thought of punishment.