It's been many years since I did any real mathematics but last night after pondering the process involved in my mathematical journey I had an idea about the abstraction of how mathematical analysis works.
As we know, mathematics is about transforming logical statements into different forms. We use a series of tautological statements to get from point A to point B and hence A and B are equivalent, logically speaking. B maybe be more appropriate/useful in some instance than A even though it is mathematically equivalent.
Therefore, suppose a sphere represents an equipotent surface where any path along the sphere represents a "equivalence" or tautological movement. That is, any two points on the sphere are tautologically equivalent.
Then the majority of pure mathematics can be seen as finding paths along the surface of the sphere. A "mathematical proof" would essentially be a closed path on the sphere.
Approximations could be seen as moving off the surface of the sphere(to a new sphere).
In fact, we would not necessarily have spheres(but this seems like it would be a topological space) but arbitrary surfaces and it would require a higher dimension than 3. (since we can approximate functions in multiple ways with each one not necessarily being equivalent yet still close to the original).
One issue of the above has is that there seems to be no real metric for "Closeness" although maybe something could be developed. e.g., given two equivalent mathematical functions or statements, say point A and B, then how close is A to B? This would be required to visualize such things in a metric space(which is initialize how I conceived of it but not necessarily how it is).
In any case, the question is about such higher meta-mathematical analysis of knowledge. The above applies to just about anything where one thing is transformed into something else through equivalence. Equivalence derivation "moves one element to another along the same "dimension"" and approximate derivations move normal to that dimension.
Is there any theories out there like this that I could read more about?