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In theory all math topics build upon more simple topics - to build a conceptual hierarchical framework.

At the bottom we may have the topic of numbers, then above that the topics of addition and subtraction, then above that multiplication and division, then somewhere above that linear algebra, etc. Each topic building upon the previous topics. A topic somewhere in the hierarchy cannot be understood without understanding the lower-level topics in which it conceptually depends.

Does such a hierarchy/framework exist? If not a hierarchy then possibly a directed dependency graph? The value of such a structure is that a student could follow a path up the hierarchy (or through a graph), knowing that if they understand all the dependencies for a given topic then they have the understanding to master it.

sebjwallace
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    The curriculum already has implemented such a "dependency graph". For example, before learning cohomology theory of groups we have linear algebra, abstract algebra, group theory and homological algebra, say. – Dietrich Burde Aug 01 '20 at 08:25
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    If you start digging into how numbers are placed in such a conceptual hierarchical framework far from its bottom with the "bottom" being set theory or topos theory, and perdicate calculus and logic even below that, you may lose your confidence in your knowing what addition, subtraction, ... is ;) – Hagen von Eitzen Aug 01 '20 at 08:26
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    @Dietrich Burde: Yes and no. The curriculum isn’t really made to understand math from the foundations – the overall movement is sonewhat top-bottom imo because the foundations are often very general, very abstract and more complex than their most natural use cases. It would be a pedagogical nightmare to study general topology and Dedekind cuts/Cauchy sequences before any concept of real numbers (and thus metric or normed spaces). Can you imagine proving rigorously that $\mathbb{C}$ is algebraically closed before letting your students play with roots of complex polynomials? – Aphelli Aug 01 '20 at 09:13
  • Anyway, my point is that making the hierarchy fully rigorous is really not an easy task. Stacks Project is a good example (ie, it would be ludicrous to use as a textbook, but very useful as a reference) for a small subset of math (algebra and algebraic geometry). – Aphelli Aug 01 '20 at 09:16
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    @Mindlack My comment was only referring to the remark "that a student could follow a path". I find it reasonable to follow a certain path when choosing the lectures. But this is not about a "conceptual hierarchy" but just a practical guide for studying (this is what the OP may need more). – Dietrich Burde Aug 01 '20 at 10:40
  • @Dietrich Burde: if the question is about a guide for study, then I misunderstood it and indeed my comment is rather irrelevant. Then yes, I suppose that existing curricula are the best approximations. – Aphelli Aug 01 '20 at 11:41
  • @Mindlack I think you understood it right. But I believe that one of the reasons for asking this question is also of practical nature (reading the last paragraph). – Dietrich Burde Aug 01 '20 at 11:50
  • Thank for for the discussions. My post did have a practical agenda. I'm working on a course where students could follow a path. I'm more convinced it's a directed graph. I'm also considering starting an open source project where mathematicians could contribute to building such a graph. This graph could act as a 'map' to help anyone on their maths journey. – sebjwallace Aug 03 '20 at 07:10

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