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."

1. "In philosophy analogies mean something very different from what they mean in mathematics. In the latter they are formulas which state the equality of two quantitative relations, and they are always constitutive; so that if three members of the proportion are given, the fourth is also given by it, that is, can be constructed out of it. In philosophy, on the contrary, the analogy is not the equality of two quantitative, but of two qualitative relations; so that if three members are given, we can know and give a priori, only the relation to a fourth, but not this fourth member itself. All I can thus gain is a rule for seeking the fourth member in experience, or a characteristic mark by which I may find it there. An analogy of experience can therefore be no more than a rule whereby unity of experience may arise from perceptions (but not telling us how perception itself, as empirical intuition in general, may arise.)"
- said Kant "in a much wordier way"

1. 'said Kant "in a much wordier way"'

:-)