Religious people often classify other people by their “religion”. It’s like each and every person must “have” one single religion, which is determined not necessarily by him/her, but rather by the religious rules.
Specifically for Judaism, the common definition used by Orthodox Jews for a “Jew” is: “A person who was born to a Jewish mother, or converted to Judaism the Orthodox way”
However, nowadays the term “Jew” is widely used for several different things. Sometimes it’s being used to describe a Jewish religious person who follows the Jewish religious rules (without getting here into the debate about different flavors of the Jewish religion). Other times it’s used to describe a person born into the Jewish “nation” (i.e., genetically, regardless of religious beliefs) and/or grew up with Jewish “culture”. Many (if not most) people in the world would probably define a “Jew” today not necessarily by religious terms.
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One of the best definitions of a “nation” was given by the late Prof. Yeshayahu Leibowitz, saying that a “nation” is a group of people referring to themselves as a “nation”. Whether one agrees with that definition or not – the Jewish “identity” (like many other identities) has been changing a lot during the years. A religious Orthodox Jew from today, if somehow appearing in old Judea, would have probably been stoned to death being considered a pagan.
Thus the Jewish identity of me and my friends is the one we grew up with: The contents about the land of Israel, the Hebrew language, the Jewish “traditions” (whether referring to actual history or legends), the holidays, the food, even the arguments and debates, in one word – the Jewish “culture”. For us, a “Jew” is not a person that has a Jewish mother, but one that grew up with the same culture and contents, loves it and wants it.
If you look for a strict mathematic definition – you won’t find it here. Somehow most people can tell who is a “French” person regardless of religion. Most people would accept the existence of “Greek” people, with their unique culture, even if they don’t worship Zeus any more. So I’m proud of my Judaism regardless of what you think about it, and I’m not a bit religious.