Halakhically, Jewish identity is determined by matrilineal descent. One is Jewish if the mother is Jewish (save conversion, of course). Whether the father is Jewish is irrelevant as to Jewish identity. R. Meir Soloveitchik explores this controversial and puzzling notion in a 2005 Azure article that I came across recently.
[Martillineal descent] seems inconsistent with the rest of Jewish law, in which it is almost always the father’s ancestry that is determinative. It is the father, and only the father, who determines a child’s status as a priest or Levite, a member of the tribe of Judah or of Benjamin, a descendant of the Hasmonean house or the Davidic. Genealogy, indeed, is determined by the father regarding all categories except the most important: Whether a child is Jewish in the first place…
The matrilineal principle is puzzling not only from the perspective of Jewish law, but from that of Jewish history as well. In The Beginnings of Jewishness, Harvard scholar Shaye Cohen points out that “throughout the ancient world the parent who mattered was, of course, the father. The children born of a marriage are his children, not the mother’s.” Aeschylus, Cohen points out, epitomized this attitude when he wrote that “The woman you call the mother of the child is not the parent; she is merely the nurse of the seed that was sown inside her.” “What, then,” asks Cohen, “are the reasons for the rabbinic matrilineal principle?”
One of the most popular explanations asserts that paternal identity is less certain than maternal identity: Since we are more likely to know who the mother of a given child is, we are best off relying on her for definitive lineage. But as Cohen observes, this explanation fails for two reasons. First, the rabbis looked to the mother’s lineage only with regard to Jewishness; if parental certainty were the central issue, then we would expect to see the matrilineal criterion for other questions of lineage. Second, the rabbis gave the mother legal standing to determine the identity of her child’s father even in cases where paternity is the defining element… [W]hen paternity is uncertain, and we rely on the mother’s testimony or location, it is never the mother’s lineage that becomes definitive. Ultimately, Cohen says, the academic historian cannot explain matrilineal descent by appealing to any ordinary historical or social factors. Though “it is easy to believe” that rabbinic Judaism, in insisting on the matrilineal principle, “must have been compelled by some societal need,” nevertheless, Cohen concludes, “there is little evidence to support this belief.”
To understand the principle of matrilineal descent, then, it is necessary to look beyond historical or sociological factors. I will propose here a theological explanation of the matrilineal principle, and show that far from being inconsistent with the rest of Jewish law, it follows from a proper understanding of the nature of Jewishness. Indeed, the principle of matrilineal descent lends insight into the Jewish view of parenthood, and even of the nature of religion itself.
R. Meir Soloveitchik proceeds to describe the rabbinic conception of motherhood and its relationship to the central motif of family in Judaism. It’s a fascinating account even you don’t ultimately accept his thesis. It’s hard to summarize or to excerpt key passages because he builds on a number of halakhic and midrashic ideas so I highly recommend reading the whole thing. Here’s his conclusion:
[B]ecause Judaism involves the election of a natural family, it is Jewish women rather than men who serve as the foundation of our familial faith. If, despite disinterest and disregard for one’s heritage, a Jew cannot sever his or her bond to nation, family, and covenant, it is because the Almighty guarantees, to paraphrase Isaiah, that a mother cannot forget her child, nor refrain from having mercy on the child she bore, and that God, therefore, will not forget Israel either. Anyone born to a Jewish mother is bound, by her motherly love, and by God’s motherly love, to the Jewish family and to every other Jew. The centrality of mother-love in Judaism thus means that all Jews are linked by familial ties that can never be undone. Born into a Judaism that is not just a faith but a family, we are all joined for eternity to God–and to each other.