There isn't a binary division of public and private, but a spectrum of various types and levels of government involvement. For example, the basic building block of what you would term private behavior--contract--requires government to exist. Absent government, contract performance is optional, which seriously impedes contracting. Even the Mogadishu arms bazaar has some government involved--the warlord in charge has a monopoly on violence and decides what the rules are going to be in the bazaar. Government is always involved; the only question is how.
Indeed, consider what the US economy would look like with no federal spending. The defense sector, the higher education sector, the transportation sector, and the agricultural sector would barely exist. We wouldn't have the Internet. The financial sector would also be in huge trouble because of lack of market confidence absent a credible regulatory regime.
My analysis starts from the assumption that government exists in every form of human society....There's so much to agree with here that it is hard to articulate a criticism, but try this: he doesn't go nearly far enough. It's not just that "government is always involved." The point is that by our very nature, we are social beings, caught in a web of our loyalties and betrayals, to govern or be governed (or perhaps better, both at once) is what it is to be human. So to talk of being "free" of "government," is incoherent: freedom is never an abstraction; it is always merely the sum total of those "restrictions" that we regard necessary or appropriate.
[This explains, by the bye, why the libertarian case is at once so plausible and so incoherent. Of course we want to be free: who wouldn't? But not one libertarian in a hundred thinks through the idea of "freedom" much beyond "I want mine."]
Adam does a commendable job of trying to incorporate some coherent principles into his definition (and to be fair, of course, he is writing no more than a blog comment, not a treatise). But I think I he falls into too common a trap: by "government," he seems to be thinking of the highly specific and particular case of a post-Medieval state system--or even more specific, the Kant/Mill duty/welfare jambalaya that underlies most of our conversations about government today.
The trouble with this approach is that this kind of "government," no matter how pervasive, is culturally specific, an artifact, and occupies only a fragment of human space or time. And humans have governed and sustained government in one form or another since they crawled out of the slime. Maybe "the state" does it; but maybe not; maybe the church, or the paterfamilias, or the army, et cetera.
"The church" is a particularly useful example, because our "separation" of church and state is in large measure rooted in a particular historical accident--the fact that neither the Pope nor the Holy Roman Emperor could trump the other in the Middle Ages. It's a model which, however attractive (and I do find it attractive) would strike most people in most places as a perverse curiosity.
"The army" may be an even more useful example. We might "privatize" (interesting choice of words) all our military functions by, say, deeding West Point over to Blackwater but we wouldn't be less "governed" thereby. Ironically, the interesting thing about the military is the number of situations in which the military has washed its hands in the actual activity of governing, finding the quotidian just too damn hard and preferring to stay in the background as a court of last resort.
I suppose Adam might say he knows all this as well or better than I and no doubt he does. But confusing "the particular,"--the culturally specific modern Weberian "monopoly of legitimate violence" with the pervasive nature of government in all times and all places can leave him defending things he doesn't want or need to defend. The immediate example would be, I think, the government (sic) role in the mortgage market. We've certainly got ourselves into a horrible cockup on that one, and straightening out will at best requrie some active, ongoing "public" involvement. But I hope he doesn't want to go back to the institutional model we operated on prior to 2008--and certainly not to think that such a model is the essence of "government." Oh and by the way, without stretching this post too far: Adam, you might want to back away from Amtrak. Like you, I'm a huge fan, but for defenders of public involvement, it is not an ace card.