I'm saying show of force makes us safer in the long run than capitulating to our enemy's demands. You seem to be disagreeing.
I am saying that the jury is still out; and, while the jury is out, we might as well try not to spend billions of dollars, have our soldiers kill, and kill others in the process if we can avoid it. I am saying that the effort should have been made, and it would have been reasonable to do so: we should not have military bases in Saudi (nor anywhere else outside the U.S.); we should not be allies with Israel (what benefit do we receive from this alliance); and, we should have at least taken the opportunity to call the Taliban's bluff on their offer to hand bin Laden over to be tried. We did none of that; we resorted to war immediately (it was not a last resort).
And it isn't vengeance, it is deterrent.
Is it a good deterrent? If we could simply eradicate the terrorists, then maybe it would be a good idea (this would not be a deterrent, it would actually be completely getting rid of the threats); however, most of the top military minds (particularly, Petraeus, Nagl, McChrystal, etc.) are of the opinion that every single time we kill a civilian we are most likely creating new enemies that want to kill us. That does not seem like a deterrent.
My kid behaves himself because he knows he will get a timeout if he doesn't, not because I've negotiated with him on what is and isn't acceptable. When he does something wrong I don't bribe him to be good, I make it clear doing the wrong thing will result in an unpleasant experience.
When you punish your child, how much money does that cost you, how many soldiers are killed, and how many others are killed? If we were talking about just dropping soldiers into Afghanistan and Pakistan to go around punching individuals in the face (and nobody would be killed), then your analogy would hold; however, there is an infinite gap between physical injury and death and, therefore, there is no analogous relation.
We don't negotiate with the al quaeda after 911, we use it to serve as a warning that attacks of that sort to anybody else that may be planning it will result in their destruction.
Yet, we are creating more enemies in doing so, which actually might make us more prone to attacks and we cannot police the entire world.
On a side note, we negotiate with north Korea and by all accounts they have only strengthen their nuclear and offensive capabilities. So there is an example of where negotiation hasn't worked, and assumiing it would with a non state terrorist organization is ludicrous.
We have never "negotiated" with North Korea; we have had a hardline stance from the beginning: no nuclear weapons. That may be talking, but it is not negotiating. Individuals that are open to negotiation are open to compromise.
Further, our stance on North Korean nuclear weapons is absurd. We have nuclear weapons; why can't North Korea have them? Hell, we dropped two atomic bombs in their neighborhood. North Korea, like every other sovereign nation, understands that the possession of nuclear weapons might make them safer from attack. Moreover, nuclear power, once in place, is cheaper and more efficient than most other power sources. Negotiations, not talks and lectures, with North Korea should look like this: we will allow you to produce both nuclear power and nuclear weapons, if you allow us (or the UNAEA) to monitor the production and to keep tabs on your arsenal. We will allow the UNAEA the same access to count our arsenal.