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

Oct. 25 2019

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.

Read more at Tablet

More about: Halakhah, Judaism, Sacrifice, Talmud

What Iran Seeks to Get from Cease-Fire Negotiations

June 20 2025

Yesterday, the Iranian foreign minister flew to Geneva to meet with European diplomats. President Trump, meanwhile, indicated that cease-fire negotiations might soon begin with Iran, which would presumably involve Tehran agreeing to make concessions regarding its nuclear program, while Washington pressures Israel to halt its military activities. According to Israeli media, Iran already began putting out feelers to the U.S. earlier this week. Aviram Bellaishe considers the purpose of these overtures:

The regime’s request to return to negotiations stems from the principle of deception and delay that has guided it for decades. Iran wants to extricate itself from a situation of total destruction of its nuclear facilities. It understands that to save the nuclear program, it must stop at a point that would allow it to return to it in the shortest possible time. So long as the negotiation process leads to halting strikes on its military capabilities and preventing the destruction of the nuclear program, and enables the transfer of enriched uranium to a safe location, it can simultaneously create the two tracks in which it specializes—a false facade of negotiations alongside a hidden nuclear race.

Read more at Jerusalem Center for Security and Foreign Affairs

More about: Iran, Israeli Security, U.S. Foreign policy