Navigate to Belief section

The Birds

‘Daf Yomi’: The Talmud’s shortest and most difficult tractate is nominally about the sacrifice of feathered animals. In fact, it sets up a number of mathematical problems that delight the rabbis in their pursuit of pure knowledge.

Adam Kirsch
October 24, 2019
Original photos: Internet Archive Book Images/Flickr; The Audobon Society
Original photos: Internet Archive Book Images/Flickr; The Audobon Society
Original photos: Internet Archive Book Images/Flickr; The Audobon Society
Original photos: Internet Archive Book Images/Flickr; The Audobon Society

Literary critic Adam Kirsch is reading a page of Talmud a day, along with Jews around the world.

The Fields Medal, the equivalent of a Nobel Prize for math, is awarded every four years to an outstanding mathematician under the age of 40. According to one tabulation, fully 25% of Fields Medals have gone to people of Jewish descent—which is remarkable, when you consider that Jews make up about 0.25% of the world’s population. Why are Jews overrepresented by a factor of 100 in the top ranks of mathematicians? It’s pure speculation, but I wonder if the answer might have something to do with Tractate Kinnim, the brief section of the Talmud that Daf Yomi readers studied last week.

“Kinnim” means “nests,” and this unusual tractate—just three pages long, consisting of all Mishna and no Gemara—is devoted to the subject of bird offerings in the Temple. While most of Seder Kodashim has been concerned with the sacrifice of large animals like sheep, there are certain occasions that require the sacrifice of birds. Notably, 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. (For this reason, the tractate consistently uses female pronouns for the person bringing the sacrifice, unlike in other tractates where he is presumptively male.)

The birds in a nest can be either pigeons or doves, but both must be of the same species. 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 to decide.

The problem that generates all the discussion in Tractate Kinnim 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. Indeed, Kinnim has a reputation as one of the most difficult parts of the Talmud, inspiring a number of very different interpretations and commentaries, because it is essentially a series of exercises in probability theory. I’ve never been very good at this kind of logic problem, and I found Kinnim hard to follow, but I’ll do my best to give a sense of its twists and turns.

The basic rule, stated in Kinnim 22b, 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: “Even if only one fledgling became intermingled with ten thousand fledglings of another type of offering, they all shall be placed in isolation until they die.” More common, however, is a situation where pairs of birds get intermingled in such a way that we know some of them are burnt offerings and some are sin offerings, but we don’t know which bird falls into which category. This is where the fun begins: How can you maximize the number of acceptable sacrifices while ensuring that no bird is sacrificed for the wrong purpose?

Say, for instance, that two women—call them Rachel and Leah, as the Koren Talmud does in its notes—are each bringing three pairs of birds to sacrifice, and they get mixed up. Now you have 12 birds, and you don’t know which belonged to which woman or were designated for which sacrifice. You might think that you could simply call six of them burnt offerings and six sin offerings. But if you did, you would run the risk that both members of one of the original pairs would both end up being offered in the same category, meaning that one of them was invalid. Instead, the rule is that you can sacrifice only six birds, three as sin offerings and three as burnt offerings, while the other six birds must be left to die of natural causes. This is the only way to make sure that you don’t end up with, say, four of Leah’s birds going for sin offerings, which would mean that one bird was definitely sacrificed incorrectly.

In Chapter 2, the rabbis look at the problem from another angle. Say that one bird from a pair brought by Leah flies away and joins a group of three pairs belonging to Rachel. You now have one solo bird and one group of seven birds. It makes sense that you can sacrifice six of those seven birds, since we know for certain that the group contains at least three burnt offerings and three sin offerings. The problem is that there is no way of knowing whether the seventh bird originally belonged to Rachel or to Leah. As a result, not only can that seventh bird not be sacrificed, neither can the one remaining bird from Leah’s original pair. That’s because of the possibility that Leah’s second bird was among the six birds sacrificed. If it was killed as a burnt offering, Leah’s remaining bird would have to go for a sin offering, and vice versa; but since there’s no way to know for sure, Leah’s solo bird can’t be sacrificed at all. And since it’s definitely consecrated, it also can’t be eaten: It has to be left to die of natural causes.

The rabbis go on to make their hypotheticals more and more complex. Say you have seven women, each with an increasing number of pairs—the first woman has two birds, the second four, the third six, and on until the seventh, who has 14. Then imagine that one bird from the first woman’s pair flew to join the second woman’s two pairs, and then one of those birds flew to the third woman’s three pairs, and so on until the last woman ended up with 15 birds. How many birds can each woman safely sacrifice? Now say that the process was reversed, so that one of the seventh woman’s 15 birds joined the sixth woman’s group, and so on until the first woman had two birds again—how many could each sacrifice now?

It’s obvious that this is not a situation that could ever 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 the 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. All fell under the category of Torah knowledge—and it was this ability to make knowledge itself sacred, I suspect, that made Judaism such a fertile source of secular intellectual achievement.


Adam Kirsch embarked on the Daf Yomi cycle of daily Talmud study in August 2012. To catch up on the complete archive, click here.

Adam Kirsch is a poet and literary critic, whose books include The People and the Books: 18 Classics of Jewish Literature.