**Non-Constructive (Disappointing) Answer:** No you don't get anything else by letting your language be two-dimensional. This is because any rigorously-defined 2D language must be *a priori* defined in the 1D language of normal mathematics/logic. That means you cannot express anything extra in the 2D language -- though you might express the same things in a more natural or insightful manner. For analogous reasons you don't get anything new by using a 3D language.

**Example:** Double categories (taken from this MathExchange post)

An object in this category is a pair of sets and a map between them. Write $(A \to B)$ for example.

Usually a morphism $f$ in a category has only one range $Dom(f)$ and codomain $Cod(f)$ assigned to it. In a double category there are two flavours of each for every morphism. A morphism in this category is a commutative square like this. . . .

$A \ \to \ B$

$\downarrow \ \ \ \ \ \ \downarrow$

$C \ \to \ D$

The 'vertical' domain and codomain are $Dom_{|} = (A \to B)$ and $Cod_| = (C \to D)$

The 'horizontal' domain and codomain are $Dom_-=(A \to C)$ and $Cod_- =(B \to D)$

There are two 'vertical' and 'horizontal' notions of composition , call them $\circ_|$ and $\circ_-$. But we can only compose two morphisms using $\circ_|$ for example if the corresponding vertical domains match up. What this means is the bottom row of the first matches the top row of the second.

We compose vertically by putting the first on top of the second to get a tall rectangle. The corners of the rectangle form a new square. That square is the vertical composition.

Likewise two squares can only be composed horizontally if the right column of the first matches the left column of the second. We compose horizontally by putting the first left of the second to get a long rectangle. The corners of the rectangle form a new square. That square is the horizontal composition.

It can occur that two morphisms may be composable horizontally but not vertically or vice versa.

Perhaps this satisfies your intuition for what a 2D version of a category should be? But observe we had no trouble defining it in normal 1D mathematics. So while the 2D perspective it might help our intuition it does not add anything in terms of the strength of the language.

**Example:** Proof Grids

Suppose $\mathcal L$ is our universe of symbols. In 1D maths a proof is a string of $N$ symbols. That means a map $\{0,1,2, \ldots, N\} \to \mathcal L$. Suppose we want a 2D proof to be a grid of $M \times M$ symbols (some symbols can be blank). That means a map $\{0,1,2, \ldots, M\}^2 \to \mathcal L$

But recall the sets $\mathbb N$ and $\mathbb N^2$ have the same cardinality.
This induces an isomorphism between the classes of 1D and 2D proofs. Thus every 2D proof can already be expressed as a 1D proof.

**Example:** Digital images

Every (two dimensional) computer image is stored as a (one dimensional) binary sequence.

"Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise."$\endgroup$ – Alex M. Mar 20 '18 at 22:29narrativefor more instrinsic underlying mathematical processes/procedures. That is, I donotthink that mathematics consists of mysterious operations on symbols,... but that symbols and operations are a way of talking about mathematics. $\endgroup$ – paul garrett Mar 20 '18 at 23:13sequenceof thoughts or statements, where each relies on the ones preceding it. Putting things like notation and the way thecontentof a (mathematical) statement is expressed aside, logic deduction itself is a fundamentally one-dimensional business. Not that I had any idea what 2-dimensional reasoning could be, though... $\endgroup$ – monkeymaths Mar 21 '18 at 10:2415more comments