TITLE: Would mathematics be different if not written one-dimensionally?
QUESTION [7 upvotes]: Mathematics is written one-dimensionally, using symbols that make sense when put together on a line. The 2d sheets of paper that we use don't have enough room to write mathematics two-dimensionally. As a result, most symbols in mathematics are operations involving two objects (or one object acting on another object) and many mathematical concepts seem to depend on this fact (non-commutativity, left/right action).

Question: Would mathematics be different for 4d beings writing on 3d sheets of paper? For example, can it make sense (for them) to extend the notion of left/right action to up/down?

REPLY [9 votes]: This should be a comment, but I do not know how to put an image there. Some commutative diagrams in homological algebra are in fact depicted "3-dimensionally" (or, at least, using perspective in order to simulate 3-dimensionality).