A lot of mathematicians consciously hold on to Mathematical Platonism. For example, Penrose in The Emperor's New Mind said there appears:
to be some profound reality about these mathematical concepts, going quite beyond the mental deliberations of any particular mathematician. It is as though human thought is, instead, being guided towards some external truth - a trith which has a reality of its own.
Others hold it unconsciously. Fir example, Eugenia Cheng has said in a paper that when she thinks about it, then she doesn't think numbers or triangles live in some Platonic realm but when she goes about her daily work as a mathematician, this is exactly how she treats it: they're there, they exist.
Now the surreal thing about mathematical platonism is that all mathematical concepts are at the same ontological level. Plato himself placed it between the matter of the world and his higher id3as/forms.
Thus for mathematical platonism, there is no difference between surreal numbers or the ordinary integers: like 1, 2, 3 ...