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>how can someone "own" a piece of Earth? or river? or mountain? or wild trees?
You are thinking of some intrinsic ownage. The right of property applies only within a society among your peers. If you own a piece of land, what it means is that you have a political sanction to use it however you like and other people within the same society can't tell you what to do on it. The right is political. If a person is all alone, rights wouldn't apply.
>Really? just because it's wild, it's first come first serve?
Yeah, really, first come first serve. The cost of exploration plus the effort it takes to get there and build an improvement earns you the right to that property among your peers.
>What about the folks who were here first? Don't they, then, really "own" the land?
What folks? The natives? Assume you are talking about the natives, they don't recognize property right in their pre-colonial collectivist societies. It was all right by might.
>Why is the goal non-contradictory justice? Why not everyone being content?
Because human begins have free will. Some people will choose unreasonable goals and won't be content unless they have their way. An Utopia is a society where everyone is happy regardless of their choices in life and related consequences. That is a physical impossibility for humans.
When people come together to form a society, their goal is to better their own lives by trading with each other in all manners. The political rules of a society for humans should be made for this goal, to designate objective rights to it's members and draw just lines on economic claims. A good political system is one that delivers objective justice which allows people to run their own lives smoothly, no more, no less.
>Aren't you a bit utopic in your idealism of capitalism?
If you mean unrealistic, no. Capitalism won't somehow magically make people happy, anyone truly defending free-market should know this. What capitalism does is deliver stern justice. You earn what you sow.