I have heard that matrilineal descent (i.e. having Judaism passed down through the mother) comes from the Roman period when women were raped by Roman soldiers and the babies were then said to be Jewish because they knew for certain who the mother was, but they did not know if the father was Jewish. Is there any truth to this claim?
Different people will argue different positions. But besides the derasha on Devarim 7:4 mentioned in Menachem's answer to this question, a straightforward counter-claim can be made from Ezra 10:2-3 (my emphasis):
ב וַיַּעַן שְׁכַנְיָה בֶן-יְחִיאֵל מִבְּנֵי עולם (עֵילָם), וַיֹּאמֶר לְעֶזְרָא--אֲנַחְנוּ מָעַלְנוּ בֵאלֹהֵינוּ, וַנֹּשֶׁב נָשִׁים נָכְרִיּוֹת מֵעַמֵּי הָאָרֶץ; וְעַתָּה יֵשׁ-מִקְוֶה לְיִשְׂרָאֵל, עַל-זֹאת
2. And Shecaniah the son of Jehiel, one of the sons of Elam, answered and said unto Ezra: 'We have broken faith with our God, and have married foreign women of the peoples of the land; yet now there is hope for Israel concerning this thing.
ג וְעַתָּה נִכְרָת-בְּרִית לֵאלֹהֵינוּ לְהוֹצִיא כָל-נָשִׁים וְהַנּוֹלָד מֵהֶם, בַּעֲצַת אֲדֹנָי, וְהַחֲרֵדִים, בְּמִצְוַת אֱלֹהֵינוּ; וְכַתּוֹרָה, יֵעָשֶׂה
3. Now therefore let us make a covenant with our God to put away all the wives, and such as are born of them, according to the counsel of the LORD, and of those that tremble at the commandment of our God; and let it be done according to the law.
See also the relevant pasuk cited in the previous perek (Ezra 9:12).
Other possible proofs (though, admittedly, not as strong as Josh's):
In Lev. 24:10ff, we have the story of "the son of an Israelite woman, who was also the son of an Egyptian man" who blasphemed Hashem's name and was executed judicially for this offense. Now it is true that, according to halachah, non-Jews are liable to death for this too (Rambam, Laws of Kings 9:3), but the specific mode of execution that was used, stoning, is used only for Jews; a non-Jew who blasphemes is beheaded (ibid. 9:14). So we see that he was considered a Jew because his mother was one.
In I Kings 7:14, Chiram, the artisan whom King Shlomo commissioned to make the copper vessels for the Beis Hamikdash, is described as "the son of a widow from the tribe of Naftali, and his father was a Tyrian, a coppersmith."
Now, most of the commentaries, noting that in II Chron. 2:13 his mother is described as "from the daughters of Dan," reconcile this by explaining that Chiram's father was indeed a Jew from the tribe of Naftali, who resided in Tyre (i.e., "Tyrian" describes his citizenship, not his nationality), and his wife (Chiram's mother) was from Dan.
However - and this is where it becomes relevant to the question of matrilineal descent - Abarbanel (and also Malbim) follow the straightforward understanding of the verse, and explain that Chiram's father was indeed a Tyrian gentile who had married a Jewish woman (and it was her parents who were, respectively, from the tribes of Naftali and Dan). Nonetheless, we see that Chiram is identified as a Jew.
א. כִּי יְבִיאֲךָ יְ־הֹוָ־ה אֱלֹהֶיךָ אֶל הָאָרֶץ אֲשֶׁר אַתָּה בָא שָׁמָּה לְרִשְׁתָּהּ וְנָשַׁל גּוֹיִם רַבִּים מִפָּנֶיךָ הַחִתִּי וְהַגִּרְגָּשִׁי וְהָאֱמֹרִי וְהַכְּנַעֲנִי וְהַפְּרִזִּי וְהַחִוִּי וְהַיְבוּסִי שִׁבְעָה גוֹיִם רַבִּים וַעֲצוּמִים מִמֶּךָּ:
ב. וּנְתָנָם יְ־הֹוָ־ה אֱלֹהֶיךָ לְפָנֶיךָ וְהִכִּיתָם הַחֲרֵם תַּחֲרִים אֹתָם לֹא תִכְרֹת לָהֶם בְּרִית וְלֹא תְחָנֵּם:
ג. וְלֹא תִתְחַתֵּן בָּם בִּתְּךָ לֹא תִתֵּן לִבְנוֹ וּבִתּוֹ לֹא תִקַּח לִבְנֶךָ:
ד. כִּי יָסִיר אֶת בִּנְךָ מֵאַחֲרַי וְעָבְדוּ אֱ־לֹהִים אֲחֵרִים וְחָרָה אַף יְ־הֹוָ־ה בָּכֶם וְהִשְׁמִידְךָ מַהֵר:
- When the Lord, your God, brings you into the land to to which you are coming to possess it, He will cast away many nations from before you: the Hittites, the Girgashites, the Amorites, the Canaanites, the Perizzites, the Hivvites, and the Jebusites, seven nations more numerous and powerful that you.
- And the Lord, your God, will deliver them to you, and you shall smite them. You shall utterly destroy them; neither shall you make a covenant with them, nor be gracious to them.
- You shall not intermarry with them; you shall not give your daughter to his son, and you shall not take his daughter for your son.
- For he will turn away your son from following Me, and they will worship the gods of others, and the wrath of the Lord will be kindled against you, and He will quickly destroy you.
Verse 4: For he will turn your son away from me.
If the son of the gentile will marry your daughter, he will turn away your son (grandson) whom your daughter will bear for him, from following after Me. From this we learn that your daughter's son, who is born of a gentile, is considered your son; but your son's son, born of a gentile woman, is not considered your son, but rather her son; for it is not said [regarding the prohibition:] "Do not take his daughter for you son," because she will turn [your (grand)son away from Me], but only that he (the gentile) will turn your son, etc.
You can see more of the same discussion here.
As an aside (since it was asked above): The Lubavitcher Rebbe once wrote in a letter (I don't have the source right now), that it makes sense logically that Jewishness goes after the mother and lineage (e.g Cohen or Levi) goes after the father:
- Since the essential existence of the baby is developed entirely by the mother in her womb, it makes sense that the essential nature of the baby (i.e. jewish or not) is determined by the nature of the mother.
- On the other hand, being a Cohen or Levi involves practical action (namely, serving in the Holy Temple), and it is the father who is obligated to teach the son. Also, the father is the one who is obligated and actually performs the service, not the mother. So the child learns from the father how to be a Cohen or Levi. Therefore it makes sense that lineage goes after the father.
Halachic sources for matrilineal descent (from Ask The Rabbi):
The Mishna in Kiddushin 66b states that if a child's mother is not Jewish, then the child is "like her," (i.e., not Jewish). This Halacha is codified in the Shulchan Aruch, Even HaEzer 8:5, without mention of any dissenting opinion. No source in the Torah teaches otherwise, and this question has never been raised in any classical Halachic text. It is an obvious and accepted axiom given to us at Sinai.