Were the rabbis mathematical geniuses? We’ve seen some fancy number work in the Talmud, but today’s page contains a mathematical calculation that was not fully understood until the late 20th century.
The mishnah toward the top of today’s page concerns cases in which three widows are owed ketubah payments of different amounts, and the estate is not large enough to pay out what is owed. How should the money then be apportioned between the three claimants? As it so often does, instead of stating the general principle, the mishnah offers only examples — leaving it to readers to figure out the underlying principle. In lieu of quoting the mishnah today (which is wordy), here’s a table that contains the rabbinic recommendations in the three specific instances described in the mishnah:
|Creditor 1 (owed 100)||Creditor 2 (owed 200)||Creditor 3 (owed 300)|
|Estate of 100||33 1/2||33 1/2||33 1/2|
|Estate of 200||50||75||75|
|Estate of 300||50||100||150|
Note that the three widows, in all cases, are owed 100, 200 and 300 dinars respectively. Therefore, 600 dinars would be required to pay them fully. Alas, in these cases, the estate is worth less than that: either 100, 200 or 300 dinars. When you first look at this chart, it might strike you that the rabbinic recommendations for dividing up the estate seem inconsistent. An estate of 100 dinars is divided equally among the widows, regardless of how much they are owed. An estate of 300 dinars, however, is divided exactly in proportion to what each widow is entitled to: in this case, half of what she is owed. In the middle example, neither of these principles seems to be at play, as the widow owed the least gets 50 dinars while the widows owed 200 and 300 respectively each get 75.
So what’s going on? Why do the rabbis recommend these particular ways of dividing the estate?
In the Talmud, the rule for dividing property between two claimants is much better understood. It comes from a parallel mishnah in Bava Metzia that describes the case of a cloak over which two people claim full or partial ownership. The first claimant says that the garment is jointly owned (in other words, she claims half the value) and the second says that it belongs entirely to her. In this case, as well as similar cases in the Tosefta (Bava Metzia 1:1) and on Yevamot 38a, the rabbis decide as follows: Any amount that one side doesn’t claim goes straight to the other side. Anything that both of them claim gets divided equally.
But this principle does not obviously explain what’s going on in today’s mishnah. For example, if the estate is worth 200, then both the second and third creditors stake a claim to all of it, and one would expect that whatever the first creditor claims must be split equally with both of them, awarding her 33 ⅓ at most — and yet she receives a full half of her claim.
The rabbis of the Gemara, confronted with the numbers in our mishnah, seem not to have been able to make heads or tails of it. They attempted to use a version of the algorithm for two claimants, and then to correct for the discrepancy between their results and those of the mishnah by making a significant limitation (an okimta) on the original case, whereby the first and second creditors have made an arrangement on the side to avoid competing over some of the estate. The okimta seems far-fetched given the simple and universal wording of the mishnah. Several medieval rabbis expressed discomfort with the Gemara’s solution, but were also unable to understand how to extend the mishnah to a general rule, so the mystery remained unsolved for over a millennium.
Then, in the 1980s, the Israeli game theorists Yisrael Aumann and Michael Maschler used some modern mathematics to answer the question in a much more satisfactory way. They pointed out that all of the rulings given by the mishnah ensure that between any two creditors, the distribution follows the garment-sharing rule. That is, if any two of the three creditors were to return their payouts to a shared pot, and then bring that pot to court along with their original respective claims, the court would award each of them exactly the amount that they had just put in. Put another way, once any of the three creditors leaves with their payout, the amount that the other two get is described precisely by the garment-sharing rule.
In other words, the mishnah is recommending a distribution by which no creditor has a case for suing any other. The fact that this distribution always exists, no matter how many creditors there are or how much of the estate they each claim, is not obvious without some clever mathematics. Nor is the fact that the distribution is unique — given all the inputs, there’s only one output that has the property we just described. Does this mean that the rabbis were secretly experts in modern game theory?
In their paper revealing this discovery, Aumann and Maschler make it clear that there is no reason to think so. They present several intuitive paths through which intelligent, numerate judges of financial law in late antiquity could have come up with this principle. Unfortunately, these important insights were ignored by countless well-meaning rabbis and educators who, throughout my childhood and early adult life as an enthusiastic mathematician, directed me to our daf as evidence of the extraordinary mathematical prowess of the mishnaic rabbis. The idea that through their study of Torah, the rabbis became anachronistically gifted in all intellectual pursuits — an extreme version of what is often called da’at torah — is put forward vaguely and tentatively in a few places in traditional rabbinic texts, but seems to have become much more popular in recent centuries in response to a Jewish community with increasing access to, and success in, the secular academic world.
This insecurity about whether Jewish wisdom can compete against Western math and science doesn’t add up for me. I have a great deal of respect for my mathematical colleagues, and I find that mathematicians tend to have excellent values and ethical intuitions. But there is no mathematical formula for describing fairness, and when settling family disputes a doctoral thesis is never as useful as generations of received wisdom and decades of sensitive listening as a communal and religious leader.
Read all of Ketubot 93 on Sefaria.