Pascal’s rules for axioms (taken from here):
- Not to omit any necessary principle without asking whether it is admitted, however clear and evident it may be.
- Not to demand, in axioms, any but things that are perfectly evident of themselves.
A number of issues seem to arise here:
- What is the status of the self-evident? Obviously it plays an important role. Is the modern conception of self-evident still the basis for axioms? What about the axiom of choice? Or are the self-evident issues not in the domain of the axioms but occur among the theorems which can be derived out of the axioms?
- How are decisions about the choice of axioms made? Are they more or less arbitrary? How is the quality of axioms to be judged? Is there any danger of begging the question here?
- Is there any discussion taking place in order to agree upon the axioms? On which ground does this discussion take place? What are the arguments? Obviously there cannot be demonstrations. Are there any historical examples of such a discourse?
- What about subfields of mathematics which are not axiomatized? What kinds of arguments are accepted there? How is agreement reached, if demonstration from first axioms is impossible? Does plain argumentation enter the scene? Probably here the self-evident items (relations between items) serve as presuppositions, as ‘pillars’ around which the rest of the ‘theory’ is ‘build’.
- Can a system of self-evident items be compared with a spanning set in linear algebra? A system of axioms will correspond in this analogy to a ‘basis’ of a vector space. A ‘basis’ of a theory can but must not necessarily be a subset of the ‘spanning set’ of self-evident items (like it is the case in a vector space).