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Rabbi Elazar ben Hisma said: The laws of mixed bird offerings (kinnim) and the key to the calculations of menstruation days (pitchei niddah) – these are the body of the halacha. The calculation of the equinoxes and geometry (gematriot) are appetizers for wisdom.-(Avot 3:18)



Rabbi Elazar ben Hisma was a mathematician who, we are told perhaps hyperbolically, could calculate the drops of the sea (Horayot 10a). In this mishna, he identifies two sets of Torah laws that rely on mathematical calculations – kinnim and pitchei niddah – that he considers “the body of halacha.” These serve in contrast to “equinoxes and geometry,” also math related, which are considered just “appetizers.” Commentators disagree about these latter disciplines. Are they meant solely as prerequisites to understanding certain Torah laws related to calculating the new moon or the numerical values of Biblical words, or are they independent secular subjects that have value as part of wisdom in their own right?

Jewish studies scholar Amram Tropper assumes that Rabbi Elazar viewed geometry and astronomy as independently important. He associates Rabbi Elazar’s approach with the Greco-Roman setting in which the Sages of the Mishna were embedded. Comparing the statement to the Greek philosopher Quintilian, Tropper argues that we can use Greco-Roman sources to better understand Rabbi Elazar’s position:

Quintilian’s claim that children should be taught geometry because it “exercises their minds, sharpens their wits and generates quickness of perception” sheds light on Rabbi Elazar’s, saying “Equinoxes and arithmetic are appetizers for wisdom.” For Quintilian, geometry and astronomy were taught to aspiring young orators for their intrinsic value and because they trained one to think logically. In a similar vein, it seems that similar disciplines were emphasized in the rabbinic curriculum because the rabbis believed, like their counterparts in rhetoric, that logical reasoning was an indispensable tool in higher education…

Rabbi Menachem haMeiri – who also believed in the importance of mathematical and philosophical knowledge – points out that the reason why Rabbi Elazar chose kinnim and pitchei niddah as his examples is because those two are the last two tractates of Mishna for Seder Kodshim and Taharot, respectively. This indicates that Rabbi Elazar encouraged learning about equinoxes and geometry only after one has a comprehensive understanding of the Torah corpus. In either case, Rabbi Elazar clearly delineates a hierarchy between the first two subjects and the latter two which, though important, are still less fundamental than what would be considered full-fledged Torah study.

Both kinnim and pitchei niddah are filled with calculations and resolution of doubtful statistical cases. In the conclusion of his book The Birth of Doubt: Confronting Uncertainty in Early Rabbinic Literature, Professor Moshe Halbertal points to kinnim as being “perhaps the starkest example in all of Tannaitic literature of developing cases of uncertainty that have nothing to do with any attempt to guide human behavior in the event that someone encounters such uncertainty in real life.” Unlike niddah calculations, which may impact real-life halacha, the doubts and calculations in kinnim are generally removed from practical ramifications. While practical halacha is important, Halbertal notes, the Sages also valued mental acuity and intellectual development through theoretical cases:

This development was motivated by the [s]ages’ fondness for legal paradoxes and unusual halachic results for which these states of uncertainty are a gold mine, and whose formulation is a sort of test of a sage’s virtuosity. In addition, the postulation of these imaginary cases is of great service for conceptual clarification and for the examination of a variety of underlying principles that inform different fields of law.

Following this understanding, Rabbi Elazar counterbalances the importance of developing both Torah and mathematical acumen, even on matters that don’t have immediate practical relevance, against the notion in the previous mishna which highlighted the importance of action over theory.

Maharal adds that the decision to highlight kinnim and pitchei niddah teaches a lesson in character growth. Rabbi Elazar’s choice of topics that relate to doubts teaches that wisdom isn’t always black and white, logical, or completely affirmative and resolute. Wisdom can be found even amidst doubt and uncertainty. With the guidance of halacha, we could live prosperously even amidst complicated uncertainty.

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Rabbi Dr. Mordechai Schiffman is an Assistant Professor at Yeshiva University’s Azrieli Graduate School, an instructor at RIETS, and the Straus Center for Torah and Western Thought. He graduated YU with a BA in psychology, an MS in Jewish Education from Azrieli and Rabbinic Ordination from RIETS, before attending St. John’s University for his doctorate in psychology.He learned for two years at Yeshivat Netiv Aryeh. He has been on the rabbinic staff of Kingsway Jewish Center in Brooklyn, NY since 2010 and practices as a licensed psychologist in NY. His book “Psyched for Torah,” his academic and popular articles, as well as many of his lectures are accessible on his website,