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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.

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    Great question. Better minds than I will have actual answers for you; I just wanted to suggest that you cross-post this to mathoverflow as well. That site is geared toward research-level mathematics, and may have other answers for you.2011-12-11
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    @Drew: I don't know if this would be within the [StackExchange policies](http://meta.stackexchange.com/questions/64068/is-cross-posting-a-question-on-multiple-stack-exchange-sites-permitted-if-the-qu). Although the argument "Each site is focussed at a specific topic area." doesn't really apply for MathSE vs. MathOverflow, I guess I'll leave it here for now. There might come good answers from MathOverflow, I agree, but I don't know what the common opinion on cross posting is. If the answers here seem not thorough, then I don't mind if the question gets shifted there.2011-12-11
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    Really good question; this is something I struggle with constantly when trying to organize the mathematics I am learning.2011-12-12
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    @DrewChristianson: Crossposting between MO and math.SE is generally [discouraged](http://tea.mathoverflow.net/discussion/1181/) mainly to avoid duplicity of efforts. Before posting on MO, let a few days of no activity pass. See e.g. [here](http://tea.mathoverflow.net/discussion/1143/). A re-post should of course link back here, if there's been any kind of answer.2011-12-12
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    I think the kind of consideration you express was at the basis of the Bourbaki project (although that was pre-category-theory of course). Notably, I think they tried to present every result in the "most general" setting in which it is valid (although one never can be sure about the existence of even more abstract and general results). Studying that case in detail will reveal both the forces and the weaknesses of the approach, and also many concrete instances of deduction from generality.2011-12-12
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    @Marc: +1. I'm aware of Bourbaki and they are indeed in the spirit of my question. Although they are eventually concerned with [the bigger picture](http://img502.imageshack.us/img502/3254/92640741.png) and give good examples, the books are somewhat didactic textbooks. I'm not necessarily concerned with learning. For example, I think that trying to understand a small differential equation helps understanding more complicated ones. (Btw. there is a hilarious introduction in one of Dieudonnés treatises, where he bashes the limit-definition of derivatives and the decrepitude of Riemann integrals.)2011-12-12
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    @NikolajKunst, about Dieudonné, see http://mathoverflow.net/questions/52708/why-should-one-still-teach-riemann-integration.2011-12-15
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    I think realization of such a project would be great only for ordering and saving our knowledge. If you try to teach someone starting from most general concepts, you will obtain only mathematical text converters not a real mathematiсians.2011-12-15
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    @lhf: Cool, I'm going to check that thread.2011-12-15
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    @Nikolaj: Could you point me where exactly does he critise limit-definition of derivatives? If not, can you tell me what is his preferred definition? In either case, thanks!2011-12-16
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    @Lovre: He is stressing the notion of the derivative as linear approximation and functional on the tangent space. He complains about the limit definition, which suggests it's just a way of getting a number out of a function (slope). Viewed this way, the derivative of a function would be again just a function. The higher dimensional generalization, where it's more clear that you have a functional, since you have to choose a direction to get a number, reduces to this, because in one dimension there is only "forward". Check "Treatise on Analysis - Vol 1: Foundations of Modern Analysis", page 147.2011-12-17
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    @Nikolaj: Thank you very much. (sorry for taking up space here in comments, if you want we can delete these comments)2011-12-17
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    [This](http://mathoverflow.net/questions/17964/is-there-a-known-way-to-formalise-notion-that-certain-theorems-are-essential-on) Math Overflow thread is somewhat related.2013-05-28

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