It seems to me that Wittgenstein was very much interested in the political goings on of his time. He was interested in the conflicts that had been causing great suffering in Europe and the world in general. I take it, his interest in Dostoevsky and his novel The Brothers Karamazov stoked this interest. The following example taken from an essay arguing that Wittgenstein was interested in the world, supports this claim.
Philosophy as a Service Industry, or, Reintroducing the Philosophical Life
ESA SAARINEN & T. P. USCHANOV, Department of Philosophy, University of Helsinki
…In the autumn of 1939, Wittgenstein and his friend Norman Malcolm were walking along the river Cam in Cambridge when they saw a newspaper vendor's sign announcing that the German government had accused the British government of instigating an attempt to assassinate Hitler. When Wittgenstein remarked that it wouldn't surprise him at all if it were true, Malcolm retorted that "the British were too civilized and decent to attempt anything so underhand, and . . . such an act was incompatible with the British 'national character'." Wittgenstein was furious, and the incident broke off his relations to Malcolm for some time (Malcolm, p. 30). Five years later, he wrote to Malcolm:
Whenever I thought of you I couldn't help thinking of a particular incident which seemed to me very important. . . . you made a remark about 'national character' that shocked me by its primitiveness. I then thought: what is the use of studying philosophy if all that it does for you is to enable you to talk with some plausibility about some abstruse questions of logic, etc., & if it does not improve your thinking about the important questions of everyday life, if it does not make you more conscientious than any . . . journalist in the use of the DANGEROUS phrases such people use for their own ends. (Malcolm, p. 93)
What is the use of studying philosophy if it doesn't improve your thinking about the important questions of everyday life? Contradicting the standard academic account of what Wittgenstein was up to, we believe that this is the pressing question he asked himself throughout his philosophical career. It was also a question Wittgenstein thought of as outweighing any specific philosophical theses or theories. But it is also exactly the question that has been forgotten and even laughed at by the mainstream of today's professional philosophy.