Having read through the above discussion, I don’t think you have distinguished between the claim that there are mathematical entities, and the claim that there are mathematical facts. The latter can mean nothing more than different mathematicians will find the same solutions to a given problem, which you accept. Call the second claim epistemological realism, and the first metaphysical realism. To argue that convergence on a set of facts can only be, or be explained by, form of metaphysical realism is to give to much credence to realism. Metaphysical realism about mathematical entities , Platonism, is much more controversial than realism about physical bodies.
Having read through the above discussion, I don’t think you have distinguished between the claim that there are mathematical entities, and the claim that there are mathematical facts. The latter can mean nothing more than different mathematicians will find the same solutions to a given problem, which you accept. Call the second claim epistemological realism, and the first metaphysical realism. To argue that convergence on a set of facts can only be, or be explained by, form of metaphysical realism is to give to much credence to realism. Metaphysical realism about mathematical entities , Platonism, is much more controversial than realism about physical bodies.