This question comes from Kishinev, from a young student there: Iael asks- What makes a Jew a Jew? How important is it for a Jew to be halachikally Jewish? If a child is born from intermarriage, why should there be a different status attributed to the child whose mother is halachically Jewish versus the one whose father is halachically Jewish? Todah!
Being Jewish according to Jewish law is not a matter of importance but a fact. If he is Jewish according to Jewish law he is a Jew with all that it entails, in terms of soul in terms of being part of the Chosen People, in terms of his role on earth. If one is not Jewish according to Jewish law he is not Jewish at all, and as any other human being he can fulfill his mission in this world by observing the Seven Laws of Noah. If he wishes to shelter under the wings of the Divine Presence and join the people of Israel, he will have to convert according to Jewish law. Under the laws of the Torah, which is the sole source of authority for the Jewish people for a few thousands years since the giving of the Torah on Mount Sinai, a Jew is anyone born to a Jewish mother and not who is born to a Jewish father. We find in the Ramban on the Torah (Leviticus 24, 10) an interesting disputation among the Rishonim whether before Matan Torah (the Giving of the Torah, at Mount Sinai) a child’s status would go according to the father as was customary among the nations, or even before Matan Torah, from after the circumcision of Avraham the child’s status is according to the mother the daughter of Avraham and is a part of the people of Israel. However, it is clear that all agree that from Matan Torah a child’s status goes only according to his mother. In Ezra (10, 2-3) we find that they called to expel from the people of Israel the foreign wives and their offspring. “And Shechaniah, the son of Jehiel, of the sons of Elam, raised his voice and said to Ezra, We have betrayed our G-d, and we have taken in foreign wives of the peoples of the land, but now there is hope for Israel concerning this. And now, let us make a covenant with our G-d to cast out all the wives and their offspring, by the counsel of the Lord and those who hasten to [perform] the commandment of our G-d, and according to the Law it shall be done”. From here it is clear that the status of the children who were born to the foreign wives go according to their mothers and not to their Jewish fathers. Therefore they were considered foreigners who should be expelled from the people of Israel. The Mishnah in Tractate Kiddushin 66b states that if a child's mother is not Jewish, then the child is not Jewish. In Talmud Tractate Kiddushin 68b, Rabbi Yochanan in the name of Rabbi Shimon Ben Yochai derives this law from the verse “For he (the gentile father) will turn away your son from following Me” (Deuteronomy 7, 4) which comes right after the verse “You shall not intermarry with them” (Deuteronomy 7, 3). Your son from a Jewish woman is called your son, and your son from a gentile woman is not called your son but her son. This law is found also in the Mishnah Yevamot 22a, and in the Talmud Yevamot 45b. The Rambam (Laws of Forbidden Relationships, 15, 3-4) rules based on the above: When a gentile or a servant enter into relations with a Jewish woman, the child is acceptable. [This applies] whether the woman is unmarried or married, whether she engaged in relations unwillingly or willingly. This is the general principle: When a child is born from a servant, a gentile, a maid-servant, or a female gentile, he is like his mother. We are not concerned with the father. This law is also stated in Shulchan Aruch Even Haezer 4, 5 & 19. Regarding the reason that only a child born to a Jewish mother is Jewish and not one born to a Jewish father, Rabbi Yechiel Yaakov Weinberg ZT”L explains: According to the laws of the Torah an offspring goes after the mother. We do not know the reasoning of the Torah. Some say it is a biological reason in the offspring’s creation and his physical and mental traits. Others say that there is a moral reason, since the mother and her constant influence are crucial for the children’s education. And this reason is implicit in the Torah, as known. Others see the reason as the undoubted certainty of the belonging of the born child to his mother. But may the reasons be whatever they may be, here the rational basis of the scientific inquiry is not decisive, but the religious basis, which was the decisive factor in forming the nation’s identity and forming the Hebrew family. Needless to say, that this law of our Torah routed itself in the soul of the nation, and affected its historical development, and the stability of the national consciousness (Liphrakim, p. 307). Beyond the biological question of who is considered Jewish there is a question no less important, what is the meaning of being a Jew and in what way is it expressed. Well known are the words of Rabbeinu Saadiah Gaon "That our nation is not a nation only with its Torah’s" (Haemunot Vehadeot, third article Chap. 7). We received the most wonderful gift from the Creator of the world. We received life. And together with life we received the "operating instructions" for life - the Holy Torah; the written Torah and the oral Torah. The meaning of being a Jew is to live right, to extract the utmost from the gift of life, to live according to the “operating instructions”. A Jew who does not learn Torah and does not observe the Torah remains, indeed, biologically Jewish "Israel even though he has sinned, is still Israel" (Sanhedrin 44a), but he is an unfortunate person who has received the most wonderful gift of life, and instead of enjoying it he ruins it, because he was not wise enough to use the attached “operating instructions”.