How much of Mathematics is limited by our writing?

Question

I'm sorry if this question is too vague or otherwise a stupid question.

Suppose the mathematicians in some alien civilisation similar to ours sculpted their Mathematics in three dimensions (or higher), rather than writing it in two, ceteris paribus. Would they have an advantage over us?

I suppose the notation would be more information-dense, meaning more nuanced ideas might be better captured than in our own myriad ways of writing Mathematics on paper. (It's fun to imagine what it might look like.)

Then again, I suppose that since the dimensions are orthogonal, such notation could be unwrapped (or "coordinatised") to become a left-to-right string like so many of us are used to. This relates to the mantra of, "if you can do it, you can do it in a Turing machine!". (I don't understand this fully, so maybe it's where the answer lies.)

We already have some two dimensional notation, like Penrose notation and Feynman diagrams. It's powerful stuff. So even on paper we can ask ourselves this:

How much of Mathematics is limited by our writing?

I started thinking about this when solving some problems in Semigroup Theory, where knowing your left from your right is extremely important. I thought, "what about up and down, back and forth?" - and pictured notation like a Dominoes game.

Answer 1

The premise is wrong. We can write 3-dimensional. Have a look at this diagram for instance. 3-dimensional diagrams appear quite often in category theory.

Answer 2

The concept of mathematical proofs is not limited by the fact that we write in two dimensions. Sure; you can't draw a 3D object properly in two dimensions, but that just helps us visualize certain structures. It doesn't change our ability to analyze them mathematically. In fact, there are many structures which cannot simply be visualized. A very basic example are Hilbert spaces which are infinite dimensional.

Answer 3

Mathematics is not limited by writing. That we 'write' (we mostly type nowadays, writing isn't the relevant characteristic of our way communication - more relevant is that we use visual ways of communication - let it be formal symbols on a paper, geometric shapes, ...), is a coincidence. (OK, maybe not a coincidence, maybe it's a consequence of our biologic/cognitive properties that we prefer writing over other options, maybe it's social, whatever the reason, it doesn't matter here).

There's no reason, a priori, why we (humans) shouldn't be able to do and communicate mathematics in totally non-visual ways. For instance strictly by sounds. (In fact, aren't staffs a $100\%$ non auditory way of transmitting music?). Science has progressed enough that human minds have been directly connected. It's probably just a matter of time that we'll be able to transmit ideas simply by thought without any resort to sense perception.

In the sense explained above, we're not limited by writing at all.

However it can be argued that our brains cannot apprehend knowledge if not by visual means. Even if we use strictly non-visual ways of communication, it could be the case that our brain translate that information into a visual language and only then can we understand it. In this way we can be limited. I don't know if this necessarily happens, I'm just saying it could happen. Better ask about this paragraph at Cognitive Science S.E..