One of the first of Nietzsche's points we came across was that science is itself rests on the assumption that truth is worth pursuing to an absolute, and that it is a fundamentally valuable pursuit. I think that Nietzsche, for all his purported radicalness (I'm tired of -isms and the dictionary says it's a word) was not actually making a particularly original point. His seems to be just another statement of the very old logical problem of resolving the statement "Truth can only be established by logical inference from known truths". The paradox is that there are essentially no truths that can be established without some "weak" assumption--hence Descartes' abortive attempts to extrapolate a logical system from "I think, therefore I am.", itself arguably a weak presupposition. The same problem is also called "bootstrapping", trying to pull oneself up by one's bootstraps; with nothing externally, verifiably "true" the axiomatic pursuit of truth is arguably useless and meaningless. Indeed, a faction of Greek philosophers known as Skeptics used a very similar argument to reject, to varying degrees, established "truths".
An amusing statement of this problem is that "it's turtles all the way down", a phrase attributed to Stephen Hawking which refers to the supposed worldview in some anonymous, primitive society that the earth is a disc supported on the back of a giant turtle, to which it is rather obvious to wonder what supports the turtle, giving the reply "it's turtles all the way down". The worldview is obviously unsatisfactory, and it closely resembles the dilemma Nietzsche pointed out. I ran across it via the hilarious Discworld series by Terry Pratchett (who is one of my favorite authors--according to the Kindle app, I've read 22 of his books) in which the eponymous disc is supported by four elephants which themselves stand on the back of a giant turtle, the Great A'tuin.
Personally, I find a lot of this debate rather amusing and not particularly deep. It seems to be more a matter of semantics than philosophy. If we give "truth" the practical definition of "consistency with all experience", which indeed is what science pursues, as well as the various arts and some philosophy in a more artsy, touchy-feely way (people act and think like so...; life feels like this...), then the problem disappears--we have an objective way of establishing truth. (It's objective, of course, only if we can take our senses, and the accounts of others of their senses, at face value. Let's all do that for the sake of convenience.)