Which parts of mathematics are objectively true or false like finite arithmetic and which are only true, false or undecidable relative to a particular axiom system such as Euclidean geometry?
This is a philosophical question that affects the practice of mathematics. For example it led to the development of alternatives to Euclidean geometry. There are well documented classical approaches including Platonism, formalism and various forms of constructivism and intuitionism. I am posting this as a community Wiki to collect less well known or alternative views and to see how they relate to my own thoughts. This is a question that can never be finally resolved given the necessary incompleteness of any sufficiently powerful mathematical system. Debatable extensions to the foundations of mathematics will always be proposed. However, it might be possible to reach consensus on an imprecise philosophical criteria for objective mathematics.
The question is taking on increasing importance today. Solomon Feferman near the end of Does mathematics need new axioms? wrote as follows.
I am convinced that the Continuum Hypothesis is an inherently vague problem that no new axiom will settle in a convincingly definite way. Moreover, I think the Platonistic philosophy of mathematics that is currently claimed to justify set theory and mathematics more generally is thoroughly unsatisfactory and that some other philosophy grounded in inter-subjective human conceptions will have to be sought to explain the apparent objectivity of mathematics.
He adds in a footnote:
CH is just the most prominent example of many set-theoretical statements that I consider to be inherently vague. Of course, one may reason confidently within set theory (e. g., in ZFC) about such statements as if they had a definite meaning.
In contrast Harvey Friedman has created a substantial body of work in which he proves certain arithmetical questions are decidable in ZFC plus certain large cardinal axioms and not decidable in ZFC or even ZFC plus smaller large cardinal axioms. Feferman's paper led to a symposium, described by Feferman in which Friedman participated.