While there are many good things to say about Libertarianism, such as its good injection of common sense into some debates, Libertarianism, as an ideology, quickly runs into some paradoxes. Perhaps not paradoxes in the strict mathematical definition of the term, but still, there are occasions when the ideology ends up taking away the very thing it promises: people's natural rights to move and do as they please.
I will present a very simple situation: the story of Farmer Jones, who owns a nice farm in the country. He is surrounded by eight other farmers, who let him carry his produce down to market every week so he can buy bread and new spades. They live in an arrangement like this:
Now, one day in our libertarian society, while Farmer Jones is sleeping, Banker Smith sneaks in and buys the eight surrounding fields for fair market value and since there is no pesky land use laws or zoning permits, puts up an entire shopping center in the middle of the night. Farmer Jones wakes up the next morning, and decides to go to town. However, Banker Smith doesn't want Hillbillies ruining the atmosphere of his brand new shopping center, so he tells Farmer Jones to stay off his land, which is well protected by uzi toting security guards (since there is no pesky gun control laws, either). Now, within our libertarian society people have a right to make decisions about their own personal property, and the government doesn't have a right to pull some eminent domain and tell Banker Smith who he can and can not let onto his land. Therefore, Banker Smith is fully within his rights to shoot Farmer Jones; and Farmer Jones very natural human right to travel as he pleases is undermined by a radical notion of property rights.
This is, of course, an extreme example of libertarian ideology. However, if you admit that it is not fair for Banker Smith to confine Farmer Jones to his patch of land until he dies, it also makes sense that Banker Smith could not ruin Farmer Jones' land by opening up hog rendering plants on the adjacent eight fields, or building heliports that frighten all of the Farmer's cows to death. And next thing you know, you have 5,000 page OSHA manuals on acceptable ergonomic chairs. Once it is accepted that libertarian ideology could strip people of their natural rights, libertarianism becomes a matter of common sense. Which sometimes it is and sometimes it isn't.
Most ideologies are flawed, and they are usually flawed for much the same reason: they have some shaky metaphysics that they won't admit to, for fear of exposing their intrinsic logical flaws or their non-relevance to the real world. Libertarian ideology comes from a metaphysics common in English philosophy, that is, the metaphysics of atoms and the void, the belief that the world can be reduced down to agents working in a neutral worldspace. According to this belief, people are agents that work in the world, and whatever they choose to do only effects them directly. To hurt another person, they must do it an agentic way: such as slipping a knife in their gut. This is the same belief system, that, when applied to economics, says that an agent or group of agents cannot influence the market conditions.
When taken to a radical extent, this metaphysics says that no agent can ever hurt another agent by interfering with the field itself: for an agent to hurt another agent, it must act directly against another agent. And yet, in our example, Banker Smith's eight fields work as eight agents to hurt Farmer Jones by depriving him of his ability to interact with the field. This is an extreme example, as examples often are, and put in a very obvious physical context: however, once the principle is understood, the same argument can be made about situations that do not involve Euler characteristics.