Let me add my my 2¢ too. I confess the issue and the arguments initially made my brain fry, it is indeed quite interesting. But this is how I view it now.
With the notational convention A[i,j] we tend to call i the row index and j the column index. Taking into consideration that Julia is column-major, it seems weird (or at least inconsistent or inconvenient) to some that the first index corresponds to a row, while it is only the second index that corresponds to a column. Perceived weirdness of this convention is even magnified when another dimension is added and we have A[i,j,k]. This may now look inconsistent because asymmetric. On one hand we say that the prioritized (fast) dimension is columns, then rows and then… how shall we call it… say, floors (with a reference to a virtual building where we put one matrix on top of another). But on the other hand we order the indices as rows, columns, floor, as in A[i,j,k]. Weird.
Or not? In fact, I no longer think so. I find it quite consistent. But I think it helps to view the i as an index with which we move along the first dimension (that is, along a given column) rather than just the index of a row. Similarly j is used to move along a given row. And k is used to move… welll, vertically (again, with the reference to a virtual building). The perceived asymetry then disappears, doesn’t it? It is not rows → columns → floors but rather indexing/moving along the given column → along the row → along… well… simply moving vertically. It seems to me perfectly consistent with column-major format.
Perhaps my displayed troubles to find a descriptive word for the third direction can partially explain where the confusion comes from. k is an index of a floor and by increasing k we are moving along… along some “vertical”. Similarly i is an index of a row but by increasing it we are moving along a column.