The theory of public goods (and bads) is pretty simple. The argument is that there are welfare-enhancing trades that would occur but do not because property rights to all the valuable characteristics of a so-called public good cannot be specified. "Basic science" is a perfect example. Presumably, we all benefit from basic science. However, scientists cannot contract with everyone who benefits from basic science. Therefore, the private benefit of basic science (to the scientist) is lower than the public benefit (to everybody) and the market will provide sub-optimal quantities of basic science. Even though the market provides basic science, the public goods theorist will be quick to point out that the market must under-provide basic science because the private return is lower than the social return.
There are three immediate problems with this line of reasoning: First, we cannot know how much of something is "optimal." That should be immediately evident. Second, one can construct an easy reductio to refute the normative implications of the theory. You, the reader, presumably benefit from the fact that I do what I can to stay healthy, I work hard to stay educated, I shower, I change my socks daily, I'm generally a nice guy, et cetera. Since I can't contract with each of you to provide a certain quantity of health, education, and good nature at a price you are willing to pay, I am a public good. It follows from the theory of public goods that I should be able to tax each of you. The ethical difficulties here should be obvious.
Third, throwing the state into the mix changes the incentive structure. As I just argued, I am a public good. Without a state, I can't tax you and society has to go on its merry little productive way. With a state, though, the route to riches may be to have oneself declared a "public good" so as to receive lots of taxpayer dollars. We might revert to a situation in which the best way to get rich is expropriation and redistribution rather than production and exchange.
We might argue (ironically) that the theory of public goods is itself a public bad, but that's another story for another day. For more reading, Tyler Cowen's edited volume The Theory of Market Failure addresses the issue in some detail, as do Hans Hoppe's writings on the private production of defense.
