The mathematics of mathematical knowledge 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?
 A: First off, I'm not altogether what you are saying, let alone what you are asking. 
You argue:

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. 

Correction/clarification: mathematics does use tautological statements to get from "point A to point B", but it also uses one directional inferences, e.g., the material conditional, to get from "point C to point D", and implications to get from D to E, hence we can infer there's a way to get from point C to E. One may not, however, "backtrack". $(C\rightarrow D \rightarrow E) \not\Longrightarrow (E \rightarrow D \rightarrow C).$ Also, $[P \land (P\rightarrow Q)] \rightarrow Q$, but we cannot then logically infer that $Q \rightarrow P$. 
So much of the rest of your argument makes little sense, given your characterization of math as consisting strictly of tautologies.
It might help if you take a look at the previous math.SE post: Common Misperceptions About Math

It seems as though you are trying to model, say, the domain of mathematical knowledge, from within that domain by using a field/concept in a sub-domain (e.g., topological space)  which emerged within the very domain you're trying to model.
What I'm trying to say that there seems to be something very "circular" about your thoughts, or analogies...
Though I am very very open to discussing the philosophy of math, perhaps more so, the philosophy of mathematical logic and practice, because I don't think enough mathematicians step "outside" of math to evaluate the assumptions they take as given.  Extraordinary work has been done in the Philosophy of Science, in this respect. Mathematics is as much about the mathematicians doing math (their assumptions as to what we take as true, their conventions which are implicitly adopted or rejected, the degree to which they believe that the work they do can be construed as purely objective, the contexts in which theories emerge, and the people "doing the thinking")... as it is about their thoughts and the products of their work.
Perhaps we need to step out of math (or rather, step into "meta-math" - perhaps higher-order math/logic) to be able to say anything about mathematical knowledge, mathematical thought, mathematical logic, and mathematical practice, rather than trying to model math from within the very conceptual domains/domain which emerged from the very model they may be trying to model.

Book suggestions:


*

*Thinking about Mathematics by Stewart Shapiro.

*Philosophy of Mathematics: Selected Readings by Paul-Benacerraf and Hilary Putnam.

A: You might be interested in reading about "topological psychology", which has aspects very similar to what you're asking about. This approach seems to have begun with Kurt Lewin's 1936 book Principles Of Topological Psychology. I remember looking at this book sometimes when I was in the library back in the 1970s as an undergraduate, but I could never figure out whether there was anything quantifiably non-trivial in the book.
Googling psychology AND topology seems to suggest that others have gone down the same road that Lewin did. See, for example, Topological Foundations of Cognitive Science by Barry Smith.
