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This is perhaps a fairly specific question, and I don't know if it belongs here or on the History stack exchange, but I would like to know more about the Rhineland academies that were headed by people like R' Yehudah ben Natan and R' Meir ben Shmuel in the 11th century, through to people like R' Moshe of Evreux and R' Eliezer of Touques in the 13th. Those schools were responsible for producing tosafot on the Babylonian Talmud: complex analyses that served to make the text an exceedingly complicated yet a consistent and self-referential system. In those academies, were students expected to have an understanding of maths or of science? Of botany? Biology? Astronomy?

If they were, where did they acquire that information? Did Jewish schools in Ashkenaz at the time provide what we today would term a "secular" scientific and mathematical education? Is there any textual evidence that confirms that the Baalei haTosafot even had a comprehension of such things, beyond what can be gleaned from the Talmudic text itself?

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Did anyone have that kind of education then? – Seth J Mar 7 at 12:55
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Without an understanding of mathematics and astronomy, Rav Saadiah Gaon could never have settled the calendrical dispute in the 10th century; without a similar understanding, Hillel II could never have formed the calendar so many years before. Rambam, in the 12th century, was extremely well-educated in a range of different subjects, as was Avraham ben Meir ibn Ezra, his contemporary. But I don't know about the sages of Ashkenaz... – Shimon bM Mar 7 at 13:21
But were they being educated the way we are, separate subjects in school with different instructors, each bearing a certification or degree in education, and covering agreed-up curricula? This is how you seem to be asking the question about the Tosafists. I think the Geonim and other sages likely were very smart people who were taught by very smart people who discussed their knowledge with each other and felt confident that what they were teaching their students was mathematically and scientifically sound. – Seth J Mar 7 at 14:11
That may have been the case with Hillel II in the 4th century, but when it came to the latter-day Geonim, to the Rambam and to Ibn Ezra, they simply went to school. Islamic schools at the time were extremely good, with a very strong focus on the sciences, and accepting of Jews. (The same was not necessarily the case if you were a Christian, nor of Christian schools in terms of their quality or their acceptance.) But in Ashkenaz Jews had to either be completely self-sufficient or travel to an Islamic land to study. Many travelled, though the sages of these academies were not known for doing so. – Shimon bM Mar 7 at 22:08

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