Top Structure

Suggestion for Top-level structure:

Basis: Proof

Supplying Mathematics

I Understanding Maths (Meta Maths)
- Questioning Job (Axioms)
- Questioning Scientific Program (Sentence)
- Questioning World (Theory)
- Questioning Universe (Formal System)
- Questioning Maths?

II Doing Maths ( Processor .... 'Homo Mathematicus' .... Scientific Community )
- Sign Processing / QA
- Clarity (Who can understand it)
- Scientific Community (Altruism)
- 'Homo Scientificus'-based Model (Interaction amongst them)
- Multi Community Model (Maths + others)

Applying Mathematics

III Understanding Application (Application to the real world (and beyond ?))
- What is the real world like?
- What Theory to choose on the Maths side?
- How to apply the Theory chosen?
- How 'good' can this approach be ( compared to others )

IV Doing Application ( The Process )
- Elements
  - Players ( Processor ... 'Homo Mathematicus'... Opportunist )
  - Interactions ( stand alone ... virtual ... restricted )
  - Areas ( Science, Engineering, ... )
- Models
  - Stand alone Models
  - Virtual (unrestricted) Communities
  - Restricted Communities
    - Restrictions in the real World
    - Models of Communities

Collection of detailed topics

Conversely - from a bottom-up view - the following terms and names should be contained in the above (distributed all over the course):

Questions:

How to write it down?
- What formal concepts of Proof do exist?
- How can I be sure that my Proof is correct?
  - Efficiency: how can I check it practically
  - Effectiveness: what can i be sure of/ how sure can I be - at all?
  - There are several ways to proof it - which one is the best?
    - Two Proofs are said to be equal iff ... (theoretical & practical answer)
    - How can I structure my Proof? (Lemmata, Corollaries & co.)
    - Are there Proof-Quality-Metrics?

The 'bright' blackbox view: Mathematician also 'generates' understanding during his work.

Questions:

Is the Proof my only result? What to do with the understanding of the Theory I have developed during my work? How can I transfer this?
- Why should I transfer it? Is this still science?
- What to transfer?
- For whom to transfer?
- How to transfer?
- Does this have an effect on the above questions?
  - Shell I seperate e.g. the question for the best way to prove sth from the transfer questions?
  - Does understanding help me with assuring the correctness?
  - Is there a notion of a formal/understandable Proof? (In practice it is, but is this just laziness or better than pure formal approaches?)

Enhanced input view: Mathematician uses already existing work as input.

Questions:

What existing Theorems may I use?
- The proved ones, but who decides this?
  - Have I to check all their proofs myself?
  - If not, who can I rely on - the scientific 'community'?

Introducing Definitions (belongs to bright blackbox view?)

Questions:

What Definitions to introduce?
- What is a Definition?
  - How is it defined?
  - How can I write Definitions down easily?
- What is it good for?
  - For formality?
  - For understanding?
- How to decide on 'define or not define'?
  - What to define always?
  - What to define never?
  - Are there good/bad examples?