Why are you only considered Jewish if your mother is?

Jewish law dictates that you are Jewish only if the woman who gave birth to you is, because when a baby is witnessed being born, you can't argue about who the mother is! However, without "100% absolute proof", there's no telling who the father could be. The law helps to ensure that regardless of the father's bloodline, since the mother is definitely Jewish, the Jewish race will continue. Judaism is one of the only religions that's also a race, so this is an important point. And since for thousands of years, someone, somewhere, is always trying to annihilate the entire Jewish population of the world, it's important to Jews that they're race, religion, & culture never be wiped off the face of the earth by anti-Semitic bigots.