Why Celebrate the Bar and Bat Mitzvah at  12 and 13?
Orthodox Perspective by Rivka C. Berman

At which exact moment does a child become an adult? Since some people never reach psychological adulthood (such as the bozos who honk and holler when they lose out on a parking spot), physical maturity was chosen as the mark of adulthood for all. Puberty generally begins in the early teen years, thus bar mitzvah age was set at thirteen years plus one day. Girls mature earlier and earn the badge of adulthood at twelve years and a day.

According to Mishna Avot 5:24, which lists a lifetime of ages and corresponding stages such as eighteen is the age for marriage, thirteen is the age for the mitzvot. Other halachic writers cite a numerical hint in the phrase “This people which I formed for me, shall recount My praise” (Isaiah 43:21). “This” in Hebrew is the word “zu” spelled with a “zayin,” whose numerical value is seven, and a “vav,” whose numerical value is six. The M’iri and Magen Avot explain this to mean thirteen year olds may be counted as part of minyan who will praise God in prayer.

Simeon and Levi were thirteen when they annihilated the city of Shechem, according to the Midrash Rabbah. In the Torah, their actions are described as “Ish Charbo,” every man took his sword. From this halachic sources surmise that since Simeon and Levi were called men at thirteen, then this is the age of male adulthood.

Since this doesn’t fully explain why thirteen and twelve are the chosen years, Torah sources credit Moses with passing on the tradition of bar/bat mitzvah age. (Rashi, Niddah 46a) There are other instances when Moses transmitted the details of Jewish practice at Sinai without jotting it down in the Torah. For example, Moses described how tefillin should look as their appearance is not described in the Torah, and Jews have been wearing uniform-looking leather boxes ever since.

Mystically speaking, the Shulchan Aruch HaRav writes that the bar/bat mitzvah birthday is the age when a child is granted a yetzer tov, a conscience, that gives them a little boost in the “be good” department.

“The consummate indwelling of the holy soul in a person takes place on the thirteenth birthday for a male and the twelfth for a female. For this reason they become responsible to fulfill the mitzvot of the Torah.”
Special emotional maturity is enough to be regarded as a Jewish adult, according to at least one halachic opinion. Specifically, a boy who lost a parent would be eligible to begin wearing tefillin and be called up to the Torah anytime after his twelfth birthday. A child who loses a parent reckons with adulthood early on. (Syrian Jews do not tend to follow this custom.)

Other cultures avoid the number thirteen. Apparently, witch covens form around the presence of twelve practitioners plus the devil, because black magic thrives on this number. Fear of the number thirteen pervades Western culture, and even has a phobia associated with it. Skyscrapers are sometimes built without a thirteenth floor. Street numbers would be set to skip the address thirteen.

Judaism regards the number thirteen differently. Thirteen denotes positive attributes in the age of a boy, in moral conduct, and in the tenets of faith in God. For example:
• A Jewish male child transitions to adulthood at Thirteen.
• God is said to have thirteen attributes of mercy. (Exodus 34:6-7)
• Maimonides, a twelfth century Torah philosopher, scholar who was a court physician on the side, ascribed thirteen principles to the Jewish faith.
 

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