It has been said that any reasonable kind of magic would be possible to study and thus, would be like just another branch of science. That is, unless it makes absolutely no sense.
I have created a setting where magic is present everywhere, but not everyone can use it. Despite millennia of effort, no one-size-fits-all system exists to describe how it works.
I want it to be impossible to really study for the common populace, otherwise my world will just end up similar to alternative history instead of having the mystique that magic brings.
The reason this magic can not be studied is that it relies on a purely mental mechanism that is individual to each person. Think of it like meditation, except that you can't trigger this mechanism through breathing exercises or any other physical activity. The effects are very much objective though. It's the classical fireball-throwing kind of magic. Casting magic requires effort in proportion to how powerful the spell is, although being skilled can reduce that effort a lot. Using magic exhausts you and once you are exhausted enough you simply can't concentrate well, just like in real life. Magic is not based on imagination, but rather is as normal to mages as moving hands is to you. It's just like a bodily function, although triggered mentally. As such, you don't need to imagine what you're about to do, but can simply do it, provided that you have learned to do it. The fatigue that using magic causes is mental. Excessive effort can cause headaches like mental work in general and you are not able to keep concentrating at your fullest all day long. The energy used for spells is out there in the world. It takes effort to control it, but you don't need to expend as much energy as the spell itself takes, since the world "pays for it". Different kinds of magic have different energy sources, but in general some kind of nature will be harmed if too much magic is used.
The technological level is akin to medieval.
Magic schools do exist, but they are not rote-of-memory like our own, since that would be basically impossible.
So here's my question: How would you study this kind of magic, if you can't study the core mechanism? How do you teach it, if it's so subjective?