By the time the novel was written, the schools weren’t rigidly classified yet. The concepts existed, they just weren’t called schools yet. But he didn’t seem to have a problem with that, no. He had a problem with Kirin Tor teaching people to think of magic as a science like math, because he thought this actively sabotages apprentices into trying to apply logic and reason into what is fundamentally paradoxical and unknowable in its true nature. You’re teaching the apprentices to reason themselves into a paradox that they fail to dig themselves out of if you insist magic must follow logic like math.
At best mages would make educated guesses based on what doesn’t work, which Khadgar demonstrates in his manual scrying spell* as he tries to uncover the secret of the Song of Aegwynn. The closer something is to working, the more catastrophic the misfire because the magic had the chance to begin to activate before something went wrong. Something that went wrong at its most fundamental level at the very beginning doesn’t even get to the point of activation, just a sad fizzle.
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The novel introduces us to the Law of Sympathy, which in essence means that you leave a little trace echo of yourself behind to everything you touch. Stronger the emotional attachment or event revolving around that item, stronger the echo. These can be used to trace the item back to its owner, or to be used as an anchor for scrying something or reliving a memory attached to it. Lacking an item like that, you must divine manually, which can be both dangerous and expensive with the reagents used. Crushed amethyst dust in particular is one known reagents. A catastrophic failure while attuning the scrying spell manually without anything to act as a guiding anchor can snap a mage’s mind.
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Mages can also use the Law of Sympathy to read letters without opening them by tapping into the echo of the writer’s intent left on the paper. This also helps them gauge their true intent if they’re lying in their text. The Law of Sympathy is thought to be an apprentice level use of magic, though most apprentices aren’t known for being creative enough yet to use it for spycraft.