That makes me wonder why they are called axes and it suggests to me that there must be some sort of 6-Dimensional space of Magic in which the various disciplines live, because that is the only reason I can think of. Using Cartesian coordinates to represent points in the space, we can give each axis a unit vector:
- Life: (1,0,0,0,0,0)
- Motion: (0,1,0,0,0,0)
- Matter: (0,0,1,0,0,0)
- Erf: (0,0,0,1,0,0)
- Fate: (0,0,0,0,1,0)
- Numbers: (0,0,0,0,0,1)
Unfortunately the subspace idea has a serious problem because some disciplines would contain other disciplines. Healomancy would be completely contained by every Fate aligned discipline, and Carnymancy would contain every Fate aligned discipline. So instead of making each discipline into a linear subspace, the disciplines could be represented by non-overlapping shapes in 6-dimensional space. There's no way of telling what those shapes would be, but surely they must be contained in the subspace for each discipline, since doing otherwise would be giving a discipline size parallel to an axis that it doesn't have. We also know that each shape is aligned to one of Erf, Fate, or Numbers, which means that it has the same distance from all points on the axis, but the shape of a discipline isn't necessarily aligned to its other axes.
Wikipedia has an article about Six-dimensional space which may offer some clues about why Magic is defined on 6 axes in Erfworld. Looking at magic this way suggests that Erfworld in total is a 9-dimensional space (at least) which strongly reminds me of String Theory, just like all the talk of Grandiocosmic Strings. Should a true fan of Erfworld understand String Theory?
I'm just guessing that all 6 dimensions of Magic are spacial dimensions because that's the sort of dimension that people usually think about, but maybe the Fate axis runs along the temporal dimension so that Erfworld actually only has 8 spacial dimensions, the 3 obvious ones, plus 5 of the magical dimensions.