School talk:Mathematics/Philosophy of Mathematics

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):


 * Aesthetics
 * Set theory
 * Logic
 * Georg Cantor & Infinity
 * Kurt G&ouml;del & Completeness & Negation-Incopleteness
 * G&ouml;del's Incompleteness Theorem
 * Continuum Hypothesis
 * Cardinality
 * Eugene Wigner & Effectiveness
 * Peano Axioms and Peano Arithmetics
 * Hermann Weyl & Cognition
 * Scepticism
 * Truth & Tarski
 * Concept & Frege

On Structuring
Questions: The 'bright' blackbox view: Mathematician also 'generates' understanding during his work.
 * 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?

Questions: Enhanced input view: Mathematician uses already existing work as input.
 * 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?)

Questions: Introducing Definitions (belongs to bright blackbox view?)
 * 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'?

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?

More Questions: ...
 * What are the properties of the Theory? (sound, complete, ...) e.g. can I find a Proof at all?
 * Is the underlying Logic (concept of Proof) the right one?

Questions concerning the scientific program: ...
 * What Theorems to approach? (esp. does it make sense to prove this Theorem at all?)
 * What is the role of this Theory amongst all other Theories?