Philosophy of mathematics/Structure

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The subject is primarly structured from the perspective of the reader, i.e. other perspectives like the ones of philosophical schools, people or history are secondary.

Since the main group adressed are people that are doing Maths or interesting in the work of people doing Maths the subject is seen from the perspective of a Mathematician that is questioning what he is doing.

The further structuring is done mainly by analysing the elements of a mathematical result and expanding the notion of the environment in which the mathematical work takes place.

Structuring According to the Elements of the Mathematical Result[edit]

This questioning takes place step-by-step from how to reasonably exploit certain degrees of freedom the Mathematician has to questioning the fundamentals required for the Mathematicians work.

Hereby the steps are structured along the elements of a mathematical result, starting with the Proof that is being delivered, the Sentence that is chosen to be proven and the underlying Theory, and finally the concept of Proof used above as well as the Language of Mathematics itself.

So 'questioning' means deciding on how to use the remaining degrees of freedom ('I have different ways to do my Proof - is one better than the others?') or getting to the bottom of things like 'my Language has some restrictions in expression - what is the cost of choosing a more expressive one?'

Thus we get the following structure:

  • Questioning the Instance of Proof
  • Questioning the Sentence
  • Questioning the Theory
  • Questioning the Concept of Proof
  • Questioning the Language

Structuring by notions of the working environment[edit]

Metamathematical results can be obtained without considering the circumstances under which the mathematical results are found.

Above this the nature of PhilM is to go beyond these purely (meta-)mathematical results, i.e. asking questions like '.' or '.'. So it expands the pure mathematical view on how results are achieved with further aspects.

These aspects are achieved by expanding the notion of the Mathematician who is doing the job. First by presuming computational costs for performing Proofs etc, then by taking human aspects like cognition and understanding into account and finally by considering the Scientific Community as a whole.

This results in questions like 'Where do Definitions make sense?' and 'How does information access influent the development of Theories?' that bring in different cognitive, social and efficiency perspectives etc.

Thus we get the following structuring:

  • Metamathematics & Computability
  • Human-Cognition & -Thinking
  • Scientific Communities and their Principals

Joining result-based and enviromental structuring[edit]

The above structuring approaches are compatible. They are joined in the following way:

  • Questioning the Instance of Proof
    • Metamathematics & Computability
    • Human-Cognition & -Thinking
    • Scientific Communities and their Principals
  • Questioning the Sentence
    • Metamathematics & Computability
    • ...
  • Questioning the Language
    • Metamathematics & Computability
    • Human-Cognition & -Thinking
    • Scientific Communities and their Principals

Although the focus of the single areas may vary. While in sections on judgement of mathematical work (using degrees of freedom) human and social issues are more in the centre of interest the focus in questioning the mathematical fundamentals is likely more on mathematical & compuntational issues.