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Regarding the Pharisees - from whom the Jewish people today are descended, as followers of both the Written Torah and Oral Torah - did believe in reincarnation, based on the Oral Law. And we believe in it today, although it is not dwelled upon in general due to our lack of understanding too much about it.
I am sending you a brief article on the subject that I was involved in writing, and I hope you find it helpful.
Reincarnation is one of the teachings of the Oral Torah. In the Written Torah there are no explicit references to reincarnation, but there are hints.
Perhaps the closest scriptual hint to this idea is Deuteronomy 25:5-10 which says that "when brothers are on the earth at the same time, and one of them dies childless, the wife of the dead brother must not marry a man outside the family. [Rather] her brother-in-law shall come to her and perform levirate marriage with her. And he shall be the first-born whom she bears; he shall succeed to the name of his dead brother, and so the name of the dead brother shall not be erased from the people of Israel... But if he refuses to marry his sister-in-law... she shall remove his shoe... His name shall be called in Israel: 'The house of him whose shoe was removed'."
The main reason for reincarnation is for the soul to fulfill its role in the creation and achieve the spiritual level for which it is destined. If a soul does not manage this in its first life, it may be given another chance, and another. If the soul did not succeed in three times, it will have to settle for whatever it has gained in the everlasting afterlife. Another reason for reincarnation is to repay a soul for its deeds in a way parallel to its sins; for example, a rich miser might be reincarnated as a poor beggar and be disregarded by a rich man, who was himself one of the paupers disregarded by the rich miser in his previous life.
Rabbi Yitzchak Luria, (the Arizal) writes that Moshe was a reincarnation of Adam's third son, Sheit (Seth), and that Sheit was a reincarnation of Hevel (Abel). (The "mem" of Moshe's name stands for "Moshe," the "shin" stands for Sheit, and the "heh" for Hevel. The great mishnaic Sage Shamai was a reincarnation of Moshe, and Hillel was a reincarnation of Aharon.
Sources:
- Zohar, Mishpatim, Exodus 1:1
- Sha'ar Hagilgulim, Hakdama 36