Fate magic is strictly speaking magic that deals with how things "are/should be".
I would have said that Fate magic is the magic of the future, because it's disciplines seem to be dominated by powers that work best when you are planning ahead. Healomancy, Dollamancy, Croakamancy are all about having units for future turns, not doing stuff immediately to the enemy. Changemancy is about equipping your troops, something one would naturally do before a battle rather than during. I'm not really satisfied with that description either, but surely that's at least some sort of clue to the true nature of Fate magic.
Numbers magic is all about how things might be. think of it as a game of statistics. If a number is involved it can be manipulated. This deals in averages and randomness. things that work in a quantum physics sort of way.
I don't see how it can be called the magic of randomness. Mathamancy seems to be the only Numbers discipline that touches upon randomness. I think that Numbers magic is about measurement and analysis. It's about figuring out the structure of the world precisely. Mathamancy, Dittomancy, and Foolamancy are all about creating simulations of real things. Consider how Foolamancy is described in B2T10
. In order to fake a thing, Jack must know what it looks like in every detail. I'd bet that Numbers magic assigns numbers to everything as it measures and quantifies the world.
Erf magic is all about the manipulation of the two. Think fractals and where you stop in zooming in.
Without being really sure what "erf" means, the fact that the world is called Erfworld leads me to believe that maybe Erf magic is the magic of manipulating the world and getting immediately powerful results. It is less concerned about the future than Fate magic and more concerned about getting things done now. While the Changemancer is busy making better weapons for some possible future battle, the Dirtamancer can be raising barracks so the troops will have a place to rest tonight
. The Erf magician doesn't analyze, like the Luckamancer doesn't calculate odds; that's for a Mathamancer.