G. E. Moore tried his hand (irresistible pun) at overcoming the Cartesian skeptical problem, that if one grants that we only experience our own mental representation of the world, knowledge (about the actual world) is impossible. In "A Defense of Common Sense" (1925) and "Proof of an External World" (1939) he proposed that we can recognize paradigm cases of knowledge (famously including "That I have a hand"), and that establishing the truth of such knowledge claims establishes the falsity of the skeptic's claim that any knowledge about the external world may be false. I find Moore's style of argument, both here and in his exposition of the "naturalistic fallacy" in Principia Ethica, to be awfully fast, but one can see how Moore felt that he was carrying on David Hume's legacy with this kind of Modernist, anti-philosophical philosophy. If there is a conflict, Hume teaches us, between what ordinary people take to be simply given and what the rationalist takes to be logically necessary, then so much the worse for logic. I'm always careful to suggest to my students that the way to think about the possible significance of Cartesian skepticism is that it shows that rationalist philosophy has problems, rather than locating the problem in ordinary belief (if the student calls me at 2AM worried that the world is not real, my counsel is that he drink some water, go for a walk, and lay off those funny cigarettes).
However, Wittgenstein zeroed in on Moore's work, which Wittgenstein felt failed to cure the philosophical disease. He wrote a short manuscript late in his life that was finally published in 1969 titled On Certainty. On Wittgenstein's view, the problem was that the skeptic used the verb "to know" in a context in which it could have no meaning. It means something to say that "He knows that Aguadilla is north of here," or "He doesn't know if there is any spaghetti sauce left," because these statements are predictive of his actions and prescriptive of ours (this is the behaviorist side of Wittgenstein); Wittgenstein develops a type of functional-role semantics. Put another way, we are genuinely informed by ordinary attributions of knowledge because it could have been different: to say "he knows" makes sense because he might not have known. The external world as such, then, is not a legitimate object of knowledge. It is part of the context in which "knowledge" is possible. Thus, we neither know nor do not know anything about the existence of the external world; the verb "to know" can gain no purchase here. Moore's strategy for defeating Cartesian skepticism fails because Moore continues to commit the same "grammatical" (as Wittgenstein calls it) error as the skeptic. Moore no more "knows" that he has a hand than the skeptic "doubts" it: both are failing to make meaningful statements.
So a question of Hume interpretation: is Hume Moore or Wittgenstein? This is significant because standard Hume interpretation holds that he was a kind of happy skeptic. The standard line is that, since he held that the boundaries of sensory experience are the boundaries of knowledge, he fully embraced the Cartesian condition, that we can never know anything other than the "impressions" that are mental, not physical things. But I think that this phenomenalist reading of Hume owes a great deal to the early analytic philosophers such as Moore and notably Russell. Hume says that knowledge has limits, and that the philosopher must be comfortable with the fact that understanding can only be extended so far and no farther, on penalty of sophistry. This strikes me as much closer to the Wittgensteinian view that "the limits of my language are the limits of my world" than the traditional interpretation of Hume, that skepticism cannot be refuted and we are left phenomenalists. Note that Wittgenstein meant that the world establishes limits for language, rather than the reverse, a common error in reading him I think. So I need now to spend some time with the Treatise, if only I had the time.
(This page was anthologized at Meaning More, thanks to them.)