> The basic question is are mathematicians fundamentally discovering mathematics (Platonism) or creating mathematics (non-Platonism)? My main issue with mathem

> The basic question is are mathematicians fundamentally discovering mathematics (Platonism) or creating mathematics (non-Platonism)? My main issue with mathematical platonics is not whether we are discovering or creating mathematic (which seems to me as semantically empty question), but whether existence of matematical object is an absolute property, or a property relative to a specific model/structure, and whether some models are metaphysically exceptional, or all models are metaphusically equal, only some are more convenient to use, so they are more worth studying. Consider some poly-platonist, who thinks that both natural-number-structure and non-standard-number structure (and both models of ZFC+CH and models of ZFC+non-CH) exist and are "real" and "tangible", and we discover internal relations in these structures as mathematical knowledge. > Or to put it another way, the abstract notion of "counting" seems to exist objectively and outside of our formal mathematics theories, and our formal theory of natural number is really trying to describe "counting" rather than create the notion of "counting" from scratch. I can say that we all have clear informal concept of what are natural numbers, just limitations of logic prevent us to formalize that. But i cannot see how such argument can be extended to set theory, where are plenty of arbitrary choices how to axiomatize that (e.g. ZFC vs. New Foundations).