Take the 2-minute tour ×  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 toda