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By structural hierarchy, I mean the mental concept in which things are 'done' in mathematics. At the top, you have mathematics itself, which is a collection of systems, like arithmetic, algebra, geometry, etc. At the bottom, you have you axioms, truths which cannot be logically questioned due to their self evident nature.

The problem is that I do not know what goes in between, and in which order. Roughly speaking, I imagine that a system incorporates a group of axioms, used to build proofs, which form mathematical tools, which are used to build the system.

The reason for asking, is that I feel that it would make my study of maths easier; If I can't achieve something in mathematics, whatever it may be, I can try to identify the problem on different levels, maybe my method is wrong, or maybe it's the wrong structure or technique, or maybe am not using the correct system be begin with, and so on.

user2901512
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  • This question will be closed soon, but if you wish to learn mathematics, there are no tricks, and your progress depends on your hard work. I would say, try not to make it like a program. – Faraad Armwood May 01 '16 at 21:24
  • Read a lot, talk to people also doing math, try to pick up ideas and techniques, keep training yourself. There is no ultimate strategy te become a good mathematician, though hard work pays of. – Mathematician 42 May 01 '16 at 21:27
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    @Faraad Armwood did you even read the question? Am not looking for a 'trick', heck I wasn't even talking about performing actual maths, it's an issue of nomenclature. – user2901512 May 01 '16 at 21:30
  • I rather like this question - one could answer it by ranking spaces or structures in terms of their complexity - e.g. a ring is an additive abelian group, or a metric space is a topological space. There must be some sort of 'structure' underpinning mathematical structures. – ahorn May 01 '16 at 21:30
  • @Mathematician yeah, I think that's good advice, though I only know so many people whom do maths and I don't think any of them could give me a hard and fast, explicit set of terms. All I've ever heard of is systems, and axioms, but nothing in between, that's why I came here :) – user2901512 May 01 '16 at 21:31
  • @ahorn Exactly, there's got be be some sort of order to the types of things us mathematics do. We do different things like building proofs, building systems, and using mathematical structures, or tools, to build said systems. – user2901512 May 01 '16 at 21:33
  • This sounds like category theory. – ahorn May 01 '16 at 21:34
  • @ahorn interesting, I never heard of it. I'll look into though :) – user2901512 May 01 '16 at 21:36

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This is a late answer, but the question is interesting, so here is my answer (sorry for my English, it may be rusted):

It turns out, there actually is a hierarchy in maths (you can't learn integrals without knowing differentiation, and no differentiation if basic concepts related to functions are not properly assimilated, and so on), and most people don't know how to represent it (hierarchical mind maps like opensource Freeplane are starting to become popular...but it's just a start).

That being said, the more complex math becomes (for example when dealing with multivariate calculus), new hierarchies must be defined (for instance, should the graphical (more generally, the phenomenal) aspect be kept apart from the analytical aspect of a mathematical object?), depending on the problem at hand (e.g. quantum theory depends strongly on analytical results, but geometrical ones are often required to explain some phenomena).

Math is a set of rules our collective minds have defined to explore logic and phenomena. By essence, it is supposed to be messy, since no one has the same definitions and the same problems to solve. Each hierarchy must then be defined by each person.

The learning aspect of mathematics becomes easily solved if while learning a set of concepts and rules, one organizes it into a hierarchy. By generalization, this is also true for any other type of learning.

What the others meant by saying "you should learn math this way or this way" is really, the hierarchy you search is up to you, but will certainly not be kept the same if someone else stumbles upon it, and makes it his/her. For learning purposes, books and methods are published, and courses are given by academics for each subject. Each time, a new hierarchy emerges, even if a common language is used to diffuse the knowledge.

You can copy someone else's hierarchy and not deviate for a while, but without motivation, a critical mind and creativity, knowledge is of no value and quickly dies in oblivion. Knowledge evolves, reorganizes, and needs us as a substrate to grow and be refined.

Vee Hua Zhi
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Nephi Temahuki
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