Constructing a Mathematics Wiki

Iteration I §

To construct a wiki for for Mathematics initially I had chosen this structure:

First iteration of wiki structure

It is difficult, in my opinion to have a strong divisions in Mathematics topics. But we may use the classification developed in zbMATH Open with the following first level in hierarchy grouping the numbered classes:

• f Foundations: 03
  • Logic, Proofs, Sets
• a Abstraction: 05-22
  • Combinatorics, Algebra, Abstract algebra, Number theory, etc.
• y Analysis: 26-49
  • Analysis in $\mathbb{R},{\mathbb{R}}^n,\mathbb{C}$
• g Geometry: 51-53
• t Topology: 54-58
• p Applications: 60-97

This can easily lead to problems, and thus must be handled and refactored in due course.

which finally resulted in:

f. "Formalism"
	- logic
	- proofs
	- sets
a. "Algebra and Abstraction"
	- N
	- Z
	- Q
	- R
		- exixtence and construction
		- algebra
		- plolynomials
	- a.matrices
	- a.polynomials
	- a.combinatorics
	- a.groups
	- a.vector spaces
	- a.number theory
y. "Analysis"
	- analysis on spaces
		- R
			- construction
			- real functions
			- real functions.single
			- real functions.multi
			- limits
		- analysis on metric spaces
			- functions between metric spaces
				- limits of functions between metric spaces
				- continuous functions between metric spaces
		- analysis on vector spaces
	- analysis on systems
		- dynamical systems
		- PDEs
g. "Geometry"
t. "Topology"
	- metric spaces
p. "Applied"
	- statistics

And it eventually lead to problems, as visible from the multiple presence of a note on R (the reals) under Algebra and analysis.

It was also difficult to write on metric spaces, the definition had to kept under topology but the definition of limit of functions etc. had to kept under analysis.

Iteration II §

But this gave me and idea to look it with a different light. The present system is as follows:

Second iteration of vault structure

We have a four level hierarchy:

• Mathematical Foundations
  • Logic, proofs
• Mathematical Structures
  • Essentially sets all “structure” defined on sets
  • All abstract stuff, no precise constructions, all from vector spaces to manifolds are here.
• Mathematical Spaces
  • All precise constructions.
  • Starting from all “number sets”: $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$
  • Their algebra, topology, analysis on them - all studied under them - as it should be.
  • All the other sets like ${\mathbb{R}}^n$ and matrices also fall here.
• Mathematical Systems
  • Essentially everything else.
  • Analytical systems: from dynamical system to PDEs
  • Algebraic systems: from polynomials to system of equations

//second iteration of wiki structure
 
- foundations
	- logic
- structures
	- sets
	- groups
	- rings
	- fields
	- vector spaces
	- vector spaces inner product
	- vector spaces normed
		- analysis
	- algebras
		- Grassmann/Exterior algebra
		- Clifford/Geometric algebra
		- Tensor algebra
	- metric spaces
		|- topology, analysis, etc.
	- topological spaces
	- topological vector spaces
	- Hilbert spaces
	- Fréchet spaces
	- Banach Spaces
	- topological manifolds
		- smooth
			- Riemannian
			- Pseudo-Riemannian
			- Symplectic
			- ...
	- category
- spaces
	- N
	- Z
	- Q
	- R
		|- algebra
		|- analysis
		|- topology
		|- geometry
	- R^3
	- R^n
		|- algebra
		|- analysis
		|- topology
		|- geometry
	- M_nxm(F) //matrices from field F
	- C
	- G^n
	- l^p
	- L^p
	- C^p(U)
	- C^n
- systems
	- polynomials
	- comninatorics
	- system of equations
	- sequences, series
	- dynamical systems
	- functional equations
	- PDEs
	- geometry
	- number theory
	- statistics

This shows the page for Mathematical Structures: