I have read Tractatus Logico Philosophicus, The Blue and Brown Books, and On Certainty. Later this summer I plan to read Philosophical Investigations, probably his most significant work.
Wittgenstein was something of an anti-philosopher. Rather than try to resolve philosophical problems, he tried to dissolve philosophical problems by showing that they were due to “bewitchment with language”–people playing a “language game” that suggested that there was a problem when there really wasn’t.
Wittgenstein’s most illustrious student was Elizabeth Anscombe, who was perhaps the greatest Catholic philosopher of the 20th century.
On Certainty is probably the most accessible one of his works… He approaches the problem of knowledge and gives some arguments against traditional forms of skepticism.
His philosophy is worth reading because it is an intellectual exercise, and he is insightful. It’s also important to be careful with language. I don’t agree with all of his positions, though many people don’t, I feel. Wittgenstein occupies a curious place in philosophical parlance. He is almost a commonly recognized authority–authors will occasionally try to argue that their view is compatible with Wittgenstein’s, or try to dispel some apparent contradiction with some argument Wittgenstein made. John Searle in an interview said that the only philosopher he really reads these days is Wittgenstein, because reading Wittgenstein is like a dialogue where you want to push back.
Wittgenstein’s thought is compatible and incompatible with Catholicism in different ways… He was raised Catholic and had a Catholic burial, but he was agnostic. He had a pretty favorable opinion of religious believers, and to some extent his philosophy of religious language is compatible with Catholic positions, though in others there would be some tension. (In a way he saw believers as playing a different language game than nonbelievers.) He was a “common language” philosopher, which dovetails nicely with the Catholic tendency toward realism, though Wittgenstein was more skeptical of constructing metaphysical systems.
It has been suggested (by Anthony Kenny) that Wittgenstein is a good way for analytic philosophers to “enter” Aquinas’s philosophy of mind, because they are both notably anti-Cartesian/Platonist, externalist, etc. (For Aquinas, because the intellect is one of the soul’s powers, and the soul is the form of the body, and we can encounter a human’s form just as we can encounter anything else’s form, there is a robust solution to the problem of other minds. Wittgenstein’s thought can be seen as similar to this, where mindedness has to do with observable, in principle public language acts. Some have read Wittgenstein as a behaviorist, but that seems to me a mistake.)
