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We learn that a Bar Mitzvah is at age 13, since it says in Bamidbar 5:6:

Tell the children of Israel: When a man (אִישׁ) or woman commits any of the sins against man to act treacherously against God, and that person is [found] guilty

and Levi, who was 13 at the time, was called a man when he attacked Shechem (Bereshit 34:25):

וַיְהִי בַיּוֹם הַשְּׁלִישִׁי בִּהְיוֹתָם כֹּאֲבִים וַיִּקְחוּ שְׁנֵי בְנֵי יַעֲקֹב שִׁמְעוֹן וְלֵוִי אֲחֵי דִינָה אִישׁ חַרְבּוֹ וַיָּבֹאוּ עַל הָעִיר בֶּטַח וַיַּהַרְגוּ כָּל זָכָר

Now it came to pass on the third day, when they were in pain, that Jacob's two sons, Simeon and Levi, Dinah's brothers, each (אִישׁ) took his sword, and they came upon the city with confidence, and they slew every male.

Rashi on Nazir 29B says that we don't find anyone under 13 called a man (אִישׁ) in the Torah, but Shimon and Levi who were 13, were called men (אִישׁ) when they attacked Shechem.

How do we know that a girl becomes Bat Mitzvah at 12 years old? Is there a passuk where we refer to a girl as a woman (אִשָּׁה), and we know that she was only 12?

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for men apparently Orach Chayim 55:5 + maybe one or two other refs nearby yeshiva.co/ask/?id=2327 may be of interest. –  barlop Apr 5 at 21:31
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up vote 7 down vote accepted

Levush (Orach Chaim 616:2) says that it is simply a matter of the Sages' grasp of human physiology: they knew that boys (usually) start developing signs of sexual maturity (specifically, two pubic hairs) at thirteen, and girls at twelve.

The Gemara (Niddah 45b) homiletically explains Gen. 2:22 (ויבן ה' אלקים) to mean that "women were given more understanding (בינה) than men," and that females therefore develop mentally (and physically - see this question) earlier than males do. (This doesn't explain why the gap is specifically a year, though. Perhaps it's because if for a boy the key date is an integral number of years, then by analogy, for a girl it has to be too - and since תפסת מרובה לא תפסת, there's no reason to assume that there's any more than one year's difference.)

The verse about Levi, by the way, is (I think) considered an asmachta, not an actual source for the law. (This is especially so, because (a) we usually don't derive precedents from pre-Sinaitic events, and (b) in this case, as Dave pointed out in the comments to that question, the physiological facts seem to have changed between Levi's time and the Talmudic era.) So there may indeed not be a parallel one using the term אשה. (Indeed, Rivkah is referred to as an אשה repeatedly in Gen. 24, and according to various opinions she was either three or fourteen years old at the time, not twelve.)

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Whenever it directly refers to Rivkah in Gen. 24, it uses "נַּעֲרָה". Most of the time it says "אשה", it is not directly referring to Rivkah. It is Avraham telling Eliezer to find a wife for Yitzchok, and Eliezer repeating the story. At that point, it could be anyone. Only in two places is it talking specifically about Rivkah, in Passuk 51 where it says "let her be a wife (אשה)" and passuk 67 and "she became his wife (אשה)". Here "אשה" is not "woman" but "wife", whereas Levi is called "אִישׁ" directly. (Although I guess one could argue that it doesn't matter whether it is "wife" or "woman") –  Menachem Jul 3 '11 at 19:16
    
@Menachem: but Eliezer says that whoever gives him water, היא האשה אשר הוכיח ה' לבן אדני - which sounds more like he's using אשה to mean "woman." –  Alex Jul 4 '11 at 3:27
    
true, but in that context (24:44) he's not talking about Rivkah specifically, he's describing the prayer he made to G-d, before Rivkah even showed up. He's just saying, whoever will do this will be the one. chabad.org/library/bible_cdo/aid/8219#v2444 –  Menachem Jul 4 '11 at 4:14
    
I could be way off base, but it seems that the Rebbe understands that the verse about Levi is more than an Asmachta, although it could be that I'm not understanding it correctly: sichosinenglish.org/books/bar-mitzvah/07.htm (First Sicha) –  Menachem Jul 7 '11 at 4:15
    
@Menachem: thanks. The Rebbe goes on to say (par. 3 in what you linked, אות ה in the original sicha) that there are two opinions on the subject: one sees the verse about Levi as the actual source of the din, the other understands the age of 13 as a Halachah Lemoshe Misinai (in which case the verse would indeed be an asmachta). –  Alex Jul 7 '11 at 14:11
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