In a nutshell, the libertarian idea starts with the most fundamental political question of all: what is the proper function of the state? If you look at the state, as an entity, you see that the defining characteristic of the state is power. In any society, the state is the ultimate, unappealable power. If the state mandates that you shall pay taxes, you shall pay taxes. If the state condemns you to death for breaking one of it’s laws, you will die. Now, what is the proper object to which the state should address that power? Following the “natural law” approach, libertarians conclude that the proper application of the power of the state ought to be confined to protecting the “life, liberty and pursuit of happiness” of the citizens of the state. Enforcing morality does not fit within that scope. From a Christian standpoint, it cannot fit within that scope, because as fallen men we are incapable of perfectly obeying the moral law, so if immorality is defined as a crime, the state would end up punishing the entire citizenry, hardly a workable political theory. In that theoretical framework, murder is banned by the state (deprives a person of life), theft is banned (deprives a citizen of their property - part of the “pursuit of happiness”), slavery is banned (liberty), etc., etc., etc… The state should not ban high fat foods like donuts (one of my own personal vices), because me freely buying donuts in no way interferes with somebody else’s life, liberty, or pursuit of happiness. Same with recreational drugs. Or prostitution. That’s not to say these things are good or OK, just that they are not the proper provence of the state.
“Now human law is framed for a number of human beings, the majority of whom are not perfect in virtue. Wherefore human laws do not forbid all vices, from which the virtuous abstain, but only the more grievous vices, from which it is possible for the majority to abstain; and chiefly those that are to the hurt of others, without the prohibition of which human society could not be maintained: thus human law prohibits murder, theft and such like.” Summa 1:2:95:2
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I answer that, Human government is derived from the Divine government, and should imitate it. Now although God is all-powerful and supremely good, nevertheless He allows certain evils to take place in the universe, which He might prevent, lest, without them, greater goods might be forfeited, or greater evils ensue. Accordingly in human government also, those who are in authority, rightly tolerate certain evils, lest certain goods be lost, or certain greater evils be incurred: thus
Augustine says (De Ordine ii, 4): “If you do away with harlots, the world will be convulsed with lust.”” Summa 2:2:10:11
Similarly, social welfare is also not properly within the scope of the state.
“In recent years the range of such intervention has vastly expanded, to the point of creating a new type of state, the so-called “Welfare State.” This has happened in some countries in order to respond better to many needs and demands, by remedying forms of poverty and deprivation unworthy of the human person. However, excesses and abuses, especially in recent years, have provoked very harsh criticisms of the Welfare State, dubbed the “Social Assistance State.” Malfunctions and defects in the Social Assistance State are the result of an inadequate understanding of the tasks proper to the State. Here again the principle of subsidiarity must be respected: a community of a higher order should not interfere in the internal life of a community of a lower order, depriving the latter of its functions, but rather should support it in case of need and help to coordinate its activity with the activities of the rest of society, always with a view to the common good.[100]
By intervening directly and depriving society of its responsibility, the Social Assistance State leads to a loss of human energies and an inordinate increase of public agencies, which are dominated more by bureaucratic ways of thinking than by concern for serving their clients, and which are accompanied by an enormous increase in spending. In fact, it would appear that needs are best understood and satisfied by people who are closest to them and who act as neighbors to those in need. It should be added that certain kinds of demands often call for a response which is not simply material but which is capable of perceiving the deeper human need. One thinks of the condition of refugees, immigrants, the elderly, the sick, and all those in circumstances which call for assistance, such as drug abusers: all these people can be helped effectively only by those who offer them genuine fraternal support, in addition to the necessary care.” - Pope John Paul II, “Centesimus Annus”, paragraph 48
I think you can make a pretty good case, as a Catholic, for a libertarian political philosophy.