The wax in the phrase "wax philosophical" is a pretty strange bird. Its wax is obviously not the ordinary definition of wax, which my dictionary summarizes as an "oily, water-resistant substance", a definition which also serves as a fair summary of other, closely related "waxes", as in earwax or beeswax.
Neither is, I think, the wax in "wax philosophical" referring to another sense of wax, as in to grow, and which I know best in reference to the Moon "waxing and waning"; it means, as best I know, that the Moon is shrinking and growing in size. So is waxing philosophical "growing philosophical"? Sounds pretty strange to me.
The truth is, I only know how to use this set phrase, and can't really break it down into its constituents. It seems fairly archaic; the philosophical isn't even in the standard canonical form of an adverb, with no ending "–ly". So I was wondering three things: What is the canonical definition of wax as its being used here? In what other ways can you wax? Finally, if wax is acting as a verb here, why is it philosophical, as an adjective, and not philosophically as an adverb?