[@Zeroth] I think that works pretty well as an outline for magic. With the comparison to math, knowing a lot of different very powerful spells shouldn't be possible. It should take some specialization to understand a higher tier spell.

Are there any magic relics or ruins? I'm not asking you to spoil the story, of course. But one of my ideas was about a researcher who was assigned by the academy to study a magical relic (perhaps a rod or puzzle box), but who failed to comprehend its secrets in time to make any difference.

It's definitely possible to do a similar sort of thing with formulae and knowledge instead, but I thought I'd see what you think.