The progress in n-category theory is largely about that. Instead of just studying objects, operations and equations, like "a group is a set g, together with multiplication + certain axioms" one studies objects, relations between objects, relations between relations etc. To study this up to level two, you visualize objects as regions in 2d space, relations between objects as lines between them and relations between relations as points where lines meet. Equations are then equalities between such pictures. If you look at those equalities as a dynamic process, which starts at time t=0 with one picture and time t=1 with the other, you get a picture in three dimensional space, which you can visualize as a movie (to be mathematically precise after the choice of a morse function you get a movie). One of the easiest examples would be the zig-zag identity. See for example
https://golem.ph.utexas.edu/category/2006/10/classical_vs_qu...
and scroll down.
One person who has completely gone overboard with that is Jacob Lurie, he has practically single handedly written down a foundational work of higher algebra and algebraic topology, which essentially precisely corresponds to what you are asking. How does mathematics look like if we weren't constrained to 2d paper to write down equations. Of course to actually write down those things you have to somehow project the equations to several essentially 1d sections or invoke very abstract methods to reason about them. Of course algebraic topology already has tools for that.
The early impetus came from Grothendieck I believe, who began to create a theory of "higher groupoids" in several thousand pages, ordinary groupoids being categories in which every morphism is an isomorphism.