I really love math. It can be an art or a science, depending on your motivations. I enjoy it as an artist; I appreciate it for its beauty. It is the single most independent form of truth. Its pulchritude is that of pure thought.
What is math?
In other fields, you cannot prove things the way you can in math. Many people don't really comprehend this because they never appreciably studied math from the proof perspective. An ignorant lawyer may think that he or she proves things in a courtroom the same way a mathematician does in a paper. Closer to the real confusion, a physicist may wantonly use the word "proof" in reference to some defense of his or her theory. Yes, mathematical physicists can prove math theorems which relate to physics, but no law of the physical universe can ever be fundamentally "proven" the way a mathematical theorem can. (Although, I could imagine a math-minded physicist using certain laws of physics as axioms and using those to logically and rigorously demonstrate that some other law must follow, but this is part of what I meant by "mathematical physics." The physicist who does this has not actually proven a theory, but rather has shown that if certain assumptions about the universe are true, then this other fact must also be true. Mathematicians use a similar trick by assuming their axioms, but they are constantly aware of the dependence on these axioms -- they never prove Pythagoras's Theorem is always true, but rather they prove that if you have these geometric axioms, then Pythagoras is correct.) You can never prove beyond all doubt that gravity works a certain way -- not the way you can prove the Prime Number Theorem.
So let me offer my own definition: Math is the study of precise and logical reasoning. It is looking at how to solve problems in a manner that is at once exact and rigorous. In order to gain precision, we must use abstract notions. In order to be as reasonable and correct as possible, we use the strict rules of logic in the form of proofs.
The two primary branches within mathematics -- applied or pure, are really overlapping with a lot of grey area between. The pure end of the spectrum is occupied by the reasoning which is as nicely abstract and widely general as possible. The applied is inhabited by those issues which more directly affect our everyday lives.
Probability Theory by R. Durrett