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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.

Git Gud
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Shaun
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  • This is related. – Shaun Dec 06 '14 at 14:34
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    This is largely a matter of cognitive science, certainly not limited to mathematical writing, and while we're at it, philosophy of language, and or philosophy of mind, where thought experiments are used to explore extremal cases. I'm afraid this is both off topic and too broad. – amWhy Dec 06 '14 at 15:06
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    I agree with @amWhy, there's nothing about this question that only makes sense if asked about the realm of mathematics. One could ask it about other fields of knowledge too. – Git Gud Dec 06 '14 at 15:15
  • One could say the same thing about this, @GitGud. There's beauty outside Mathematics. Should we close that question? – Shaun Dec 06 '14 at 15:16
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    @Shaun: That is not to say the question isn't interesting. – amWhy Dec 06 '14 at 15:17
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    @Shaun I don't know. In fact I'll contradict myself now. This can be asked about other areas of science, but it can be asked about mathematics too which makes it not completely off topic. I would let the question be. The community can close it if it wishes so. Edit: If you wish to delete it, I'll let you know I have no problem with my answer going to waste. – Git Gud Dec 06 '14 at 15:20
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    If we could answer your question, Shaun, it would presuppose knowledge and experience alien to our own. To know what we're missing presumes at least some among us know and experience what you describe as alien, hence no longer being alien (at least to those who can answer). If we are by nature limited to two-dimension writing, then we de facto can't know what we're missing. – amWhy Dec 06 '14 at 15:28

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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..

Community
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Git Gud
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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.

Martin Brandenburg
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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.

Daniel Goldman
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  • Thank you. I get the impression that this somehow misses the point but I can't articulate why exactly yet. Maybe I'm just being picky though. Let me think about it :) – Shaun Dec 06 '14 at 14:42
  • Sure; you can't draw a 3D object properly in two dimensions Ah: There's the problem. I'm not asking about mathematical objects with some sense of dimension, but rather the notation we use to think about mathematics, regardless of conceptual notions of dimension. – Shaun Dec 06 '14 at 14:46