as well as the inverse of something to narrow in scope that no applications for that knowledge can be devised not even a question to be answered (Fermat's Theorems being examples of such questions to be answered), are equally bad outcomes.
Again, being so focused on application is an engineers perspective and not a researcher's. At it's heart, research has always been about knowing more about the universe, that is all the purpose you need. Paleontology barely has any purpose if you believe otherwise. And my point with the Legendre polynomials was that years down the line, people might find it useful to answer questions in their own fields. As long as it isn't blatantly wrong (the Alzheimer's scam) any science is good science. And in some cases, we've had completely wrong theories still result in good applications lol (for example the initial hypothesis behind people using baking powder as a preworkout)
As someone who works in astrophysics, I keep getting asked what application my research could possibly have in our daily life, and it always pisses me off.
I was trying to find this comic that one of my old professors had stuck on the door to his office that goes like 'Admin: why should we fund this experiment?
What you have to say: because it'll help constrain xxx value to be used in yyy
What you want to say: BECAUSE IT'S FUCKING COOL!'
Can't find it so just pretend I linked it here :/
P.S. I'd have added string theory and QFT in higher dimensions or whatever that thing is called among blatantly wrong science but I didn't study it enough to know its issues