What is mathematics?
Mathematics is the study of relationships where the purely prepositional character of the relationship in question is abstracted and isolated from any other qualities any actual objects in such a relationship might have.
For example, an element is in a set or not in it: we are looking at pure "in-ness" itself, and ignoring the various ways one thing can be in another: the Thunder are in the NBA conference finals in a different way than Connecticut is in the Union, and both are different from how the dog is in the doghouse. But in set theory we don't concern ourselves with how the Thunder played to get in the finals, or whether Connecticut has a right to leave the Union, or if the dog enjoys his house. We abstract away from all that, and just consider "in" or "out"... "greater than" or "less than," "isotopic to" or "not isotopic to," "countable" or "not countable," and so on.
As Gauss said, "Mathematics is concerned only with the enumeration and comparison of relations."