In phil of math we "usually" speak of platonism, while in phil of other branches of science we prefer the term realism, which seems more "palatable" and less "ontological committed".
Basically, what the "common sense" of the scientific community share is some sort of belief about the existence of an external reality which is the reference for scientific language and theories.
Trivially, newtonian mechanics assume the existence of the planet, of an attractive force called gravitation, and so on. The same for relativity or biology.
Of course we have some "issue" here : see quantum mechanics, but the scientific community devotes a lot of time and money searching some empirical support to the existence of (e.g.) Higgs' boson. This presuppose some sort of belief in the "reality" of Higss' boson.
Assuming this very "rough" introductory discourse, my point of view is that it is natural for a mathematician to share a "common sense" belief in some sort of realism regarding the mathematical language.
In other words, I think that the natural point of view of the mathematicians regarding the language of mathematical theories is that it must have "reference".
And this is the source of troubles : where are numbers ? where are sets ?
The platonist characterization of this sort of issues comes from the fact that is "difficult" to support the "common sense" realist view regarding abstract entities.
But, and this is the point of view shared by some "distinguished philosophers which have studied the philosophy of mathematics, like Frege and Russell, and currently debated under the headings of Naturalism and Indispensability argument, can we "make sense" of all mathematical science only considering it some sort of "formal game" deprived of any reference ?