The Mathematics of Lineage

The Kuzari and the genealogies

The Torah is full of lists of “who begot whom.” Why does this matter to us?

Rabbi Yehuda Halevi, in the Kuzari (I:95), answers: there is a quality in man, “the divine matter” (inyan Elohi), a spiritual power that allows one to reach high levels. This quality is passed down by heredity from father to son. Sometimes it skips a generation: Terach did not receive it, but his son Abraham received it from his grandfather. The genealogies trace the path of this quality, until it reached the whole people of Israel.

Our question: does this idea fit with mathematics?

Malthus and the population explosion

In 1066, William the Conqueror conquered England and conducted a census to collect taxes. Thanks to him, we know how many people lived there. By 1800 the population grew from about one million to about ten million: slow growth, about 3.5% per generation. After 1800, thanks to falling mortality and improved hygiene, growth jumped to about 25% per generation. Land was divided among too many heirs, poor peasants flocked to the cities, and the tension between rich and poor shaped the 19th century.

The economist Thomas Malthus predicted disaster: there would not be enough food for everyone, and famine, war and epidemics would follow. So he opposed aid to the poor (“the poor laws create the poor which they maintain”) and recommended not marrying. To this day the prediction has not come true: wealth has grown faster than the population, and each person today earns on average almost twenty times more than two hundred years ago. Whether Malthus will be right in the end remains an open question, considered one of the great questions of science. Some economists think growth will end. Others think that more people means more ideas, and therefore more solutions.

And what does halakha say about refraining from having children?

Be fruitful and multiply: the “fast track” and the “slow track”

How many children are required? Halakha rules: a son and a daughter. It also rules that a man who passed the age of twenty without marrying would be compelled by the court to marry. But the Rema (Even HaEzer 1:3) writes that in our times we no longer compel.

Why? The Mordechai, a medieval commentator, explains from the Gemara (Bava Batra 60b). Rabbi Yishmael ben Elisha, who lived after the destruction of the Second Temple, said: since the Romans have been imposing harsh decrees on us, it would have been fitting for us to decide not to marry or have children, “and the seed of Abraham our father would come to an end of itself.” But it is better not to tell this to the people. According to the Mordechai, since it would have been fitting to cancel the mitzvah, at least we do not compel. This is the “fast track”: complete cancellation of the mitzvah, and within a generation the people disappears.

Tosafot did not accept canceling a Torah mitzvah. In their view, Rabbi Yishmael meant only that after a son and a daughter, people should not go on having children. This is the “slow track”: if every couple has only two children, the people will slowly shrink, because some of the children will not raise families. Worldwide, an average of about 2.33 children per woman is needed for the population not to shrink (in developed countries, where almost all children survive, about 2.1 is enough).

But there is a problem here: to have a son and a daughter, two children are not always enough.

The calculation: three children

Let us start with a simple question: if a couple has children until the first son and then stops, how many children will there be on average? Half the families will have a son right away, a quarter after one daughter, an eighth after two daughters, and so on. The average comes out to exactly two children.

A family that wants a son and a daughter: the first child fulfills half the requirement, and then they wait for a child of the other sex. That is the same question, so another two children on average. In total, three. And since three is more than 2.33, even on the “slow track” the people will not disappear, but grow.

In practice, most Jews over the generations did not even reach a son and a daughter. Disease, persecution and assimilation kept the people at an almost constant size. And this makes sense: if every couple had four children, within 50 generations there would be 100 people on every square meter of the earth. Genetic studies also show that throughout history, each couple had on average only slightly more than two surviving children. The Torah’s inheritance laws also suit only a population that hardly grows. The Torah makes sure that every field stays in the family forever, for example through the Jubilee and the redemption of land. But if every father has several sons, the field is divided among them in every generation, and within a few generations only tiny plots remain.

Today, medicine allows more children to survive, but the Western way of life makes families smaller. In Europe, more people already die than are born. And yet Israel is the only wealthy country where more than 2.33 children are born per woman. The Jewish people will survive, contrary to the recommendation of Rabbi Yishmael.

Who wants to be the Mashiach? “Strong” lineage

According to the prophet Isaiah (11), the Mashiach will be “a shoot from the stump of Jesse,” that is, a descendant of Jesse, the father of King David. Suppose the Mashiach must descend “son after son”: the son of the son of the son of Jesse, without any woman in between. It is not clear to what extent this is grounded in halakha. Some have suggested that the Mashiach inherits the kingship from David; but an inheritance can also pass through a daughter when there is no son.

If the Mashiach must descend “son after son,” it is like a family name passed from father to son. If only Jesse bore the name “Beit HaLachmi,” how many people bear it today?

In the 19th century the scientist Francis Galton, Darwin’s cousin, noticed that the family names of the British aristocracy were disappearing. Explanations were offered: the nobles do not want to divide their property, stress harms fertility, and even that they eat too much meat. But Galton and the clergyman Watson found a simple explanation: chance. Every man has a random number of sons, and a name that disappears does not come back. They proved that even in a stable population, almost all family names disappear over time, until only one remains. The name here is only a marker of descent: it passes only from father to son, just like the Y chromosome in our genes. So when the name disappears, the “son after son” line has been broken, even if the descent continues through the daughters.

About 100 to 150 generations have passed since David. Assuming the population hardly grew, as was the case for most of history, the chance that his “son after son” line survived is about 1 in 100 or 1 in 150. And if it did survive, it has about 100 to 150 such descendants, because the few lines that survive take the place of all those that disappeared. These are the candidates.

Why does a “son after son” line disappear? In a stable population, each man has on average one son. But that is only an average: one has no sons, another has three. So the number of descendants in a line goes up and down by chance, like a coin tossed again and again. Sooner or later there will be a generation in which none of them has sons. Once that happens, the line has ended forever: there is no way back from zero.

The puzzle of the Kohanim. The priesthood passes exactly like a family name: only from father to son. According to tradition, all Kohanim are “son after son” descendants of Aaron the Kohen. Genetic studies found a common marker on the Y chromosome, the part of our genes that passes only from father to son, in about half of all Kohanim. Among other Jews this marker is rare: under 5%. According to Galton and Watson’s calculation, such a line should almost have disappeared, or, if it survived, left a few hundred descendants at most. Yet in fact Kohanim make up about 5% of the Jewish people: tens of thousands already two hundred years ago. How is this possible? Perhaps in groups that kept close family ties, such as Kohanim who married daughters of Kohanim, more sons were born and survived. If so, the seed of David too, as a family of distinguished lineage, could have survived with a higher probability. The question remains open.

And among the Shiites. Shiite Muslims have a similar problem. They believe their leader must be a “son after son” descendant of Ali, Muhammad’s son-in-law. Most of them hold that the line was broken more than a thousand years ago. Only one stream claims that its leader, the Aga Khan, is such a descendant, but the list has some suspicious gaps.

“Weak” lineage: we are all descendants of David

And what if lineage passes through mothers too? Each of us has two parents, four grandparents, eight great-grandparents, and so on. Thirty generations ago we had about a billion ancestors, more than all the people alive at the time! This means that almost everyone who lived then is our ancestor.

Scientists have proved it: after a fairly small number of generations, each person falls into one of two cases. Either he has no descendants at all, or the whole population is descended from him. In a people of one million, this happens within about 35 generations, about 875 years. For all of humanity, within about 1,300 years. Not even mountains, seas or differences between peoples stop it.

Why are we all descended from everyone? If a person has descendants, their number roughly doubles each generation: 2, 4, 8, 16… After 10 generations they are about a thousand, and after 20 generations about a million. Very quickly their number reaches the size of the entire population. But that is not yet everyone, because many descendants are “counted twice”: cousins marry each other. So a few more generations are needed until the last ones come in too. In a people of one million, it takes about 35 generations in total.

An example: in 1701 the British decided that only the descendants of Princess Sophia of Hanover, through sons or daughters, could inherit the crown. Since the right passes to all children, the number of eligible heirs roughly doubles each generation. After about 12 generations there are already about 5,000 eligible heirs, and within a few dozen generations almost all Britons will be on the list.

So what does this tell us?

About the Kuzari. According to halakha, the son of a Jewish mother is Jewish, and the son of a Jewish father and a non-Jewish mother is not Jewish. But according to the Kuzari, the quality passes specifically through the father. If so, every time a Jew had a son with a non-Jewish woman, the quality left the people. Over the generations, most of it “leaked” out. And if the quality passes through the mother too, it already belongs to everyone.

About egalitarian priesthood. In egalitarian minyanim of the Conservative movement, both Kohanim and “Kohanot,” daughters of Kohanim, go up to recite the Priestly Blessing, so that there is equality between men and women.

But the idea is not consistent. If one truly wants equality, it is not enough for a Kohen’s daughter to bless. She must also pass the priesthood on to her children, exactly like a Kohen’s son. Then the priesthood turns from a “strong” lineage into a “weak” one, and the mathematics is clear: within a few generations almost the whole people will be Kohanim. And when everyone is a Kohen, the priesthood no longer distinguishes anyone. Full equality in effect cancels the priesthood itself.

About mamzerut. A mamzer may not marry an ordinary Jew, only a mamzeret. In mamzerut, the defect passes through both the father and the mother (Kiddushin 67b). Therefore, if even a single mamzer mixed into the people many generations ago, we are almost certainly all his descendants, meaning we are all mamzerim. Hence a surprising “permission”: a known mamzer may marry anyone, since everyone is a mamzer, and a mamzer is permitted to a mamzeret.

Some poskim answered that the Torah forbids only a certain mamzer, not someone about whom there is only a doubt. And a family whose defect was forgotten within the people remains fit forever (in the words of the Gemara: “a family that has become assimilated, has become assimilated,” Kiddushin 71a). Therefore anyone not known to be a mamzer is not a mamzer, even if statistics say otherwise.

But the Rema (Even HaEzer, end of chapter 4) implies otherwise. He forbade marrying Karaites (a sect that broke away from rabbinic Judaism), because there are mamzerim among them. Yet about each individual Karaite there is no certainty that he is a mamzer, only a concern. And still the Rema forbids, without saying “a doubtful mamzer is permitted” or “a family that has become assimilated, has become assimilated.” It follows that in his view, a concern of mamzerut mixed into an entire group is enough to forbid.

About Jesus. Christians agree that the Messiah must be a descendant of David. That is why the New Testament opens with a “son after son” list from David to Joseph, the father of Jesus. But according to their belief, Joseph was not his biological father. Abarbanel objects: if so, Jesus is not of the seed of David! The Christian answer: Mary too was of the house of David. Abarbanel (on Isaiah 11) rejects this as well. And mathematically, lineage through the mother is “weak”: after 42 generations, almost every Jew of that generation was “of the house of David.”

The same question arises regarding those who claim that the Lubavitcher Rebbe is the Mashiach: his genealogy back to David passes through a woman, the daughter of the author of the Tanya.

About the Mashiach. The Gemara (Sanhedrin 98b) says that Rabbi Yehuda HaNasi was worthy of being the Mashiach, and he was a descendant of David through daughters. If such lineage is enough, each of us is a candidate. All that remains is to polish our buttons.