In the Talmud’s Most Difficult Tractate, the Rabbis Tackle Mathematical Probability

October 25, 2019 | Adam Kirsch
About the author: Adam Kirsch, a poet and literary critic, is the author of, among other books, Benjamin Disraeli and The People and The Books: Eighteen Classics of Jewish Literature.

The talmudic tractate of Kinnim (“nests”) takes up only three pages in standard editions, and includes only Mishnah (the Talmud’s earlier stratum), without Gemara (the later stratum). As the title implies, it is concerned with bird sacrifices, which were mandated—during the time the Temple stood—for a handful of occasions. Traditionally, Kinnim is considered one of the most challenging parts of the Talmud, as Adam Kirsch explains:

[A] woman who has given birth but can’t afford to sacrifice an animal can bring a pair of birds, known collectively as a “nest,” instead. . . . The two birds are sacrificed in different ways: one is a burnt offering, which means that its blood is sprinkled on the lower half of the altar, and the other is a sin offering, whose blood is sprinkled on the upper half. The woman bringing the sacrifice can either designate which bird is for which purpose or she can leave them undesignated, so that it’s up to the priest [performing the sacrifice] to decide.

The problem . . . is that unlike sheep, which generally stay in a pen or a stall when you put them there, birds can fly. This means that it is easy for different groups of birds to get mixed up with each other: a bird from one woman’s pair could fly over and join another woman’s pair. In that case, the priest who has to sacrifice them is faced with a problem: how does he know which bird is intended for which sacrifice? What if he accidentally sacrifices a burnt offering as a sin offering or vice-versa, rendering them invalid?

This sounds like a practical problem, and no doubt it sometimes happened in the Temple that groups of birds got mixed up. But in tractate Kinnim, this simple premise seems to have been seized upon by some mathematically inclined rabbis as an excuse for inventing math and logic puzzles. . . . The basic rule, . . . is that the priest must avoid even the smallest risk of performing an invalid sacrifice by offering a bird that has been designated for a sin offering as a burnt offering, or vice-versa. . . . How can you maximize the number of acceptable sacrifices while ensuring that no bird is sacrificed for the wrong purpose? . . . The rabbis go on to make their hypotheticals more and more complex.

It’s obvious that [the thorniest of these hypotheticals] could never arise in real life—even before you take into account the fact that, in talmudic times, sacrifice was no longer practiced at all. It is a pure logic problem, a way of thinking about probabilities that delighted rabbis centuries before the invention of modern probability theory. But for the rabbis themselves, there was no clear separation between such mathematical challenges and the other matters discussed in the Talmud, from Shabbat observance to marriage and divorce law.

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