Mathematics is the study of mathematical objects.
Mathematical objects are objects rooted in a scheme of concrete manipulations (on drawings, symbols...) but view through the idealization that these manipulations can be repeated indefinitely (like a drawing a line, or adding 1 to a number).
In this way, mathematical objects are the ideal closure of their schemes of manipulations. Not purely ideal, not purely empirical.
The way I see it, mathematics is about
figuring out relations between mathematical objects (theorems) and
finding out which manipulations on some objects of study (leading to formal definitions of these objects) are effective to construct (demonstration) their expected relations.
Mathematics are about objects, not statements.
They are about constructions, not truth.
The objectivity of mathematics—like any objectivity—lies in the possibility for anybody to repeat mathematical constructions.