I started developing a game system that had three distinct types of magic.
Laymen, of course, refer to it all generally as "magic," but the learned realize that there are three completely different disciplines.
Familiar magic (or Sorcery) requires linking oneself to a magical being (fey, demon, outworlder, etc) which imbues you with power and teaches you higher knowledge. The abilities of this magic depend on the familiar: some might be good at ensnaring the mind, others at summoning bolts of fire and ice. It is possible to link to multiple familiars, but that would be like trying to hold down two or three jobs at once (or taking more than 18 hours of classes). All the familiars would demand your attention, your affection, and maybe even a measure of recompense.
Arithmancy is the magic of manipulating space and time. The language of the universe is in numbers, and arithmancers can manipulate the numbers to reshape the world around them. They can fold space to "teleport" themselves or summon things to them. They can slow time down or speed it up...or at least make things appear to do so relative to the rest of the world. They can even cross points in one world with another. Some few read into the recursiveness of numbers to foretell events of the future. Arithmancers leave nothing to chance, for numbers do not lie and are never wrong.
Alchemy is the science of transformation and transmutation. By divining the substances that make up all physical things, an alchemist can disperse and recombine those elements to create whatever he wishes. Healing the body and making lead into gold are childishly simple tasks to him. He can make make caverns into castles and moles into manticores. However, he realizes that nothing can be created ex nihilo...there must always be a balance: heat for cold, strength for weakness, life for death.
I always wished there was some way to make this idea work in 4e. I never figured out a way. Any ideas?