I tend to not like A Mathematician’s Lament for being so lofty, but there's a more straightforward reason to care about "pure" math: students complain that math

I tend to not like A Mathematician’s Lament for being so lofty, but there's a more straightforward reason to care about "pure" math: students complain that math "doesn't make sense" or is a bunch of random difficult to remember rules, which is absurd. More than any other field, math has logical reasons for everything. You can figure out pretty much anything you'll encounter in school by just thinking about it; you don't need to have any external knowledge from having done some experiment or being told some fact. On the flip side, if you do use physical intuition, you'll usually go down the right path (for anything pre-university, or even most undergraduate level material). Regardless of how beautiful music is, teaching how to read sheet music without actually listening to music is stupid because you're not teaching what the symbols even mean. Of course no one will understand what you're talking about.