In Israel, more men study in religious academies than do in university. The ones at yeshiva or kollel, schools for unmarried and married men respectively, have some idea what is taught at universities. The ones at university have no idea what's being taught at the Jewish academies.
What actually are yeshiva students doing? They are trying to understand the will of God.
God falls silent
According to religious belief, prophecy ceased to exist at the very beginning of the First Temple period, with the death of the last prophets Haggai and Malachi (or, as some believe, Ezra). Since then, God has been silent, and Jews have had to rely on the laws he had already handed down to us through revelation.
God’s revelations are manifested in the written law: the Torah and the rest of the Bible, and the Oral Law, which religious Jews believe God gave to Moses together with the Torah atop Mount Sinai.
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This Oral Law is believed to contain the correct interpretation of the Torah as well as supplemental laws and instruction, without which the Torah would be incomprehensible.
For example, the Torah says: "Thou shalt kill of thy herd and of thy flock, which the Lord hath given thee, as I have commanded thee” (Deuteronomy 12:21), but nowhere does it elucidate how exactly this kosher slaughter is to be performed. The manner of slaughter is believed to have been handed down to the Jews through the Oral Law.
According to tradition, God's words to Moses were orally transmitted onwards from generation to generation: “Moses received the Torah from Sinai and transmitted it to Joshua, and Joshua to the Elders, and the Elders to the Prophets, and the Prophets transmitted it to the Men of the Great Assembly” (Mishnah Avot 1:1).
But then the Romans destroyed the Temple in 70 C.E. and the Great Assembly was no more. Transmission of the Oral Law fell to the rabbis.
Each rabbi taught the law and his interpretation of it to his students, who went on to teach it and their own interpretation to their students, who taught it to theirs, and so on.
In the 2nd century, following the abortive Bar Kokhba revolt, the Romans laid waste to much of Jewish Palestine. A rabbi called Judah the Prince who remained in the ruined land collated the different teachings.
Judah the Prince is credited with redacting the Mishnah, a legal code arranged by tractates, each dealing with a different aspect of Jewish life, from how to sacrifice goats at the Temple to the proceedings of a legal dispute over the border of a wheat field. The Mishnah also quotes rabbis debating these laws over the generations.
Despite its complexity, the Mishnah was not written down. The tractates were memorized by Rabbi Judah the Prince’s students, who, starting in the early 3rd century C.E., continued the tradition of oral dissemination.
The Talmud, born in argument
Unlike the Bible, the Mishnah was not considered to be the "final product". For roughly the next 200 years, rabbis, now in two centers – northern Palestine and Babylonia - debated the laws, elaborating and adding to them as they went along.
These discussions and comments were then memorized by students who would become rabbis and make their own comments, and so on.
This continued until roughly the 5th century, when this giant collection of comments and discussions on the Mishnah and Jewish Law in general was redacted, giving rise to the Talmud (or rather the two Talmuds: one Palestinian and one Babylonian).
This Talmudic knowledge is referred to as the "Oral Law." And it was indeed oral, at least at first.
The Talmud was not put into writing, but was transmitted unchanged from generation to generation by word of mouth, until apparently the 8th century. From roughly that point, commenting on the Talmud – now a written text – began again.
The Talmud needed more comment and elucidation because the exact meaning of the rabbis was not always clear, if only because it is mostly in Aramaic, a language no longer commonly used by Jews.
Oral Law studied in yeshiva is, therefore, the aggregate of commentaries on Jewish Law, the Mishnah; with its commentary, the Talmud; with its commentaries; and the commentaries on these commentaries, built up over about 2,000 years. And that is what they study in yeshiva.
To this day the Oral Law remains a dynamic, growing thing, expanding and unfolding as generations of Jews study the text.
The lulav laws
So, study of Jewish Law in yeshiva is not based directly on the Bible but on the Talmud, since the rabbis of the Talmud are believed to have had a special sensitivity to the written word of God and an ability to extract whole laws from minute details of the text in ways that later generations could not.
Yeshiva students are not studying the Jewish laws per se; the laws have long been extracted from the Talmud and organized in legal codices such as the Shulkhan Arukh. To find out what a particular law is, they can look in the literature.
What Jewish academies are doing is analyzing the Talmud and its commentaries in the hope of reverse-engineering from them the underlying reason governing the Oral Law itself. By doing this, they believe that they are uncovering the laws underlying God's will.
Their employ rigorous logic to pursue their analyses. One method, for example, is seeking after seeming contradictions, that cannot be such, as the will of God cannot contradict itself.
For example, the Talmud teaches that one cannot fulfil a mitzvah by way of sin. An oft-repeated example is that stealing a lulav (date frond) to shake on Sukkot as is commanded does not fulfill the mitzvah, since stealing is a sin. But the Jerusalem Talmud also forbids tearing one's clothing on Shabbat – yet doing that in mourning for a relative does constitute fulfillment of the mitzvah of mourning.
Since this could not be a contradiction, the rabbis uncovered the underlying distinction: the sin only matters (regarding mitzvot) if it concerns the essence of the mitzvah, not something that is incidental to it.
That explains why blowing a stolen shofar on Rosh Hashanah does fulfill the mitzvah of shofar-blowing on the holiday: it is not the shofar that is essential for the mitzvah, but its sound.
God and the Large Hadron Collider
From reasoning such as this, more and more abstract categories and distinctions are created, and are used in the study of the Oral Law.
These theoretical abstractions are the goal of yeshiva learning. They are held to be more valuable than the particular rules of Jewish Law, for they uncover the underlying architecture of God’s plan for the Jews and the world.
Theoretical physicists do much the same when they conceptualize particles and processes in order to explain the underlying workings of the universe based on observation.
The difference lies in the observations being analyzed and studied. Physicists pore over data gleaned from experiments in the Large Hadron Collider. Yeshiva students pore over the Oral Law, hoping to uncover its underlying principles as a way to understand the mind of the creator of the universe.