If half of your genes come from your father, and half of your genes come from your mother, then why is it that only the mother's genes determine whether or not one is jewish, and not the fathers? I do not understand, at least scientifically, why one would not be just as jewish for having a jewish father and a catholic mother, as one would be if he had a jewish mother and a catholic father. Within Judaism, this is just how "Jewishness" is defined. It isn't based upon science or genetics, but rather upon religious belief.
Can one not be half-jewish?
From the standpoint of ethnicity, certainly. One can be half-Jewish just as one can be half-German or half-Italian. The difference is that being Jewish is more than just an ethnicity.