Various mathematical areas of research evolved from a wide and diverse range of questions. Many are physical in nature or come from informatics/computer science, some are procedural or optimization problems and so on. Often patterns emerge and lead to studies of abstract structures and in this way extend our knowledge of what is considered pure mathematics. Often, the original context gets stripped off and eventually, in the process of generalization, formerly different frameworks can be seen under a new perspective. Under some circumstances notation might be introduced and new and sometimes more elegant proofs are discovered.
If I have abstractly proven a statement, say for example in category theory (like the Grothendieck–Hirzebruch–Riemann–Roch theorem), and I then choose one of several specific frameworks, which satisfies certain structural patterns, then I can draw conclusions from the more abstract theorem and state new ones as a lemma (like the classical Riemann-Roch theorem). There are many statements in diverse geometric theories (like Poisson, Symplectic, Riemann or Euclidean geometriy, Lie Group theory etc.), which can be viewed as really just the same theorem under different lights. If I prove Stokes theorem, I might avoid finding individual proofs for various kinds of integral theorems. The core of some statements in de Rham cohomology is already hidden in more abstract cohomology theories with less specific context. My question is ultimately concerned with such a nesting and is kind of an optimization problem.
From the point of view of organization, one might consider drawing top-down conclusions more efficient than finding several more specific proofs. Lets assume one is interested in the bulk of major mathematical research areas and also able to draw all drawable conclusions for certain given axioms. It then seems to me that the following order of analysis seems like an suitable one: Introducing the notion of an abstract set, introducting all the various logics in an intelligent order, introduce abstract algebra followed by the category theoretic framework and then various more specific sets with axioms which make up relevant structures (including topology, numbers, modules, vector spaces, geometries,...). All the time with some intelligent order in mind, i.e. jumping from special case to special case in a minimalist way, even if there might not be an optimal way to do so, of course. Encyclopedic web pages like Wikipedia, MathWorld or comprehensive treatises give a rough conception of how this may work. (Personally, there are a couple of ambiguities for me, especially regarding the points at which I just have to introduce set theory. For example, I don't know if you really have to introduce sentences as objects in propositional calculus or what other big areas are explicable in non-set theoretic terms, even if they usually are explained that way. I also have little insight in modern model theoretic perspective.) Having said that:
What would be an (or maybe the) efficient order to introduce the mathematical frameworks (of current knowledge, state of the art and relevance), and why? Here the term "efficient" is understood in the sense explained above.
I would not be surprised if there are different good ways to approach this. But in any case, there are obviously theorems which include others and since one can put many mathematical disciplines on their own feet, there is some unambitious pyramiding. A more synthesizing formulation of the above question, one stated in terms of the specific fields, would be to ask:
Which are major mathematical theorems, which are strongly including in this sense? Which are theorems which have an unnegligible power with regard to special cases or sub-theorems and would therefore be worth of being proven first? As another approach, a start might be to find out which axioms are really necessary, i.e. doing reverse mathematics.