Back to Basics
In a higher category, with at least some non-identity $n$-arrows for $n \geq 2$, we still interpret diagrams as "equations". As an exercise, draw a square composed of four smaller squares, each containing a 2-arrow. Now label objects, 1-arrows and 2-arrows, and write out all the 1-dimensional paths as linear equations. Clearly, insisting on writing out equations this way would get tiresome very quickly.
If we came across a physical or mathematical structure indexed by three ordinal indices, as opposed to the two indices of matrices or numbers, we would naturally guess that equations in these objects would be 2-dimensional. These objects might be triangle 2-arrows in a 2-category whose objects are again the ordinals, but this could only be useful if the 2-arrows satisfied the right pasting conditions to form equations.